{"id":55918,"date":"2023-01-05T18:24:46","date_gmt":"2023-01-05T16:24:46","guid":{"rendered":"https:\/\/omathimatikos.gr\/?p=55918"},"modified":"2023-11-04T10:38:33","modified_gmt":"2023-11-04T08:38:33","slug":"%ce%b3%ce%b9%ce%b1%cf%84%ce%af-%cf%84%ce%bf-%ce%b7%ce%bc%ce%af%cf%84%ce%bf%ce%bd%ce%bf-%ce%ba%ce%b1%ce%b9-%cf%84%ce%bf-%cf%83%cf%85%ce%bd%ce%b7%ce%bc%ce%af%cf%84%ce%bf%ce%bd%ce%bf-%ce%ba%ce%ac%ce%bd","status":"publish","type":"post","link":"https:\/\/omathimatikos.gr\/?p=55918","title":{"rendered":"\u0393\u03b9\u03b1\u03c4\u03af \u03c4\u03bf \u03b7\u03bc\u03af\u03c4\u03bf\u03bd\u03bf (\u03ba\u03b1\u03b9 \u03c4\u03bf \u03c3\u03c5\u03bd\u03b7\u03bc\u03af\u03c4\u03bf\u03bd\u03bf) \u03ba\u03ac\u03bd\u03bf\u03c5\u03bd \u03ba\u03cd\u03bc\u03b1\u03c4\u03b1"},"content":{"rendered":"<p><!--more--><\/p>\n<p>\u03a3\u03ba\u03b5\u03c6\u03c4\u03b5\u03af\u03c4\u03b5 \u03ad\u03bd\u03b1 \u03bf\u03c1\u03b8\u03bf\u03b3\u03ce\u03bd\u03b9\u03bf \u03c4\u03c1\u03af\u03b3\u03c9\u03bd\u03bf. \u03a4\u03cc\u03c4\u03b5 \u03c0\u03b9\u03b8\u03b1\u03bd\u03cc\u03c4\u03b1\u03c4\u03b1 \u03b8\u03b1 \u03b8\u03c5\u03bc\u03ac\u03c3\u03c4\u03b5 \u03b1\u03c0\u03cc \u03c4\u03bf \u03c3\u03c7\u03bf\u03bb\u03b5\u03af\u03bf \u03cc\u03c4\u03b9 \u03bc\u03c0\u03bf\u03c1\u03b5\u03af\u03c4\u03b5 \u03bd\u03b1 \u03c7\u03c1\u03b7\u03c3\u03b9\u03bc\u03bf\u03c0\u03bf\u03b9\u03ae\u03c3\u03b5\u03c4\u03b5 \u03c4\u03b9\u03c2 \u03c3\u03c5\u03bd\u03b1\u03c1\u03c4\u03ae\u03c3\u03b5\u03b9\u03c2 \u03b7\u03bc\u03b9\u03c4\u03bf\u03bd\u03bf\u03b5\u03b9\u03b4\u03bf\u03cd\u03c2 \u03ba\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b7\u03bc\u03b9\u03c4\u03bf\u03bd\u03bf\u03b5\u03b9\u03b4\u03bf\u03cd\u03c2 \u03b3\u03b9\u03b1 \u03bd\u03b1 \u03bc\u03ac\u03b8\u03b5\u03c4\u03b5 \u03c0\u03b5\u03c1\u03b9\u03c3\u03c3\u03cc\u03c4\u03b5\u03c1\u03b1 \u03b3\u03b9\u03b1 \u03c4\u03bf \u03c4\u03c1\u03af\u03b3\u03c9\u03bd\u03bf. \u0391\u03bd\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/be978ae895058c72f35dfd22afcf4ed7\/images\/img-0001.png?w=700&#038;ssl=1\" alt=\"$\\\u03ac\u03bb\u03c6\u03b1 $\" data-recalc-dims=\"1\" \/>\u03b5\u03af\u03bd\u03b1\u03b9 \u03bc\u03af\u03b1 \u03b1\u03c0\u03cc \u03c4\u03b9\u03c2 \u03b3\u03c9\u03bd\u03af\u03b5\u03c2 \u03c0\u03bf\u03c5 \u03b4\u03b5\u03bd \u03b5\u03af\u03bd\u03b1\u03b9 \u03b7 \u03c3\u03c9\u03c3\u03c4\u03ae \u03b3\u03c9\u03bd\u03af\u03b1, \u03c4\u03cc\u03c4\u03b5 \u03ad\u03c7\u03b5\u03c4\u03b5<\/p>\n<table id=\"a0000000002\" class=\"equation\" width=\"100%\" cellspacing=\"0\" cellpadding=\"7\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/be978ae895058c72f35dfd22afcf4ed7\/images\/img-0002.png?w=700&#038;ssl=1\" alt=\"\\[ \\sin (\\alpha ) = \\frac{\\mbox{\u03bc\u03ae\u03ba\u03bf\u03c2 \u03b1\u03bd\u03c4\u03af\u03b8\u03b5\u03c4\u03b7\u03c2 \u03c0\u03bb\u03b5\u03c5\u03c1\u03ac\u03c2}}{\\mbox{\u03bc\u03ae\u03ba\u03bf\u03c2 \u03c5\u03c0\u03bf\u03c4\u03b5\u03af\u03bd\u03bf\u03c5\u03c3\u03b1\u03c2}} \\]\" data-recalc-dims=\"1\" \/><\/td>\n<td><\/td>\n<td class=\"eqnnum\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u03ba\u03b1\u03b9<\/p>\n<table id=\"a0000000003\" class=\"equation\" width=\"100%\" cellspacing=\"0\" cellpadding=\"7\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/be978ae895058c72f35dfd22afcf4ed7\/images\/img-0003.png?w=700&#038;ssl=1\" alt=\"\\[ \\cos (\\alpha ) =\\frac{\\mbox{\u03bc\u03ae\u03ba\u03bf\u03c2 \u03b4\u03b9\u03c0\u03bb\u03b1\u03bd\u03ae\u03c2 \u03c0\u03bb\u03b5\u03c5\u03c1\u03ac\u03c2}}{\\mbox{\u03bc\u03ae\u03ba\u03bf\u03c2 \u03c5\u03c0\u03bf\u03c4\u03b5\u03af\u03bd\u03bf\u03c5\u03c3\u03b1\u03c2}}. \\]\" data-recalc-dims=\"1\" \/><\/td>\n<td><\/td>\n<td class=\"eqnnum\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<div class=\"centreimage\">\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/content\/sites\/plus.maths.org\/files\/articles\/2022\/HYD2\/1280px-Trigono_sine_en2.png?resize=400%2C400&#038;ssl=1\" alt=\"\" width=\"400\" height=\"400\" data-recalc-dims=\"1\" \/><\/p>\n<p>\u0393\u03b9\u03b1 \u03c4\u03b7 \u03b3\u03c9\u03bd\u03af\u03b1 \u03b1, \u03c4\u03bf \u03b7\u03bc\u03af\u03c4\u03bf\u03bd\u03bf \u03b4\u03af\u03bd\u03b5\u03b9 \u03c4\u03bf\u03bd \u03bb\u03cc\u03b3\u03bf \u03c4\u03bf\u03c5 \u03bc\u03ae\u03ba\u03bf\u03c5\u03c2 \u03c4\u03b7\u03c2 \u03b1\u03c0\u03ad\u03bd\u03b1\u03bd\u03c4\u03b9 \u03c0\u03bb\u03b5\u03c5\u03c1\u03ac\u03c2 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03bf \u03bc\u03ae\u03ba\u03bf\u03c2 \u03c4\u03b7\u03c2 \u03c5\u03c0\u03bf\u03c4\u03b5\u03af\u03bd\u03bf\u03c5\u03c3\u03b1\u03c2. \u03a4\u03bf \u03c3\u03c5\u03bd\u03b7\u03bc\u03af\u03c4\u03bf\u03bd\u03bf \u03b4\u03af\u03bd\u03b5\u03b9 \u03c4\u03bf\u03bd \u03bb\u03cc\u03b3\u03bf \u03c4\u03bf\u03c5 \u03bc\u03ae\u03ba\u03bf\u03c5\u03c2 \u03c4\u03b7\u03c2 \u03b4\u03b9\u03c0\u03bb\u03b1\u03bd\u03ae\u03c2 \u03c0\u03bb\u03b5\u03c5\u03c1\u03ac\u03c2 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03bf \u03bc\u03ae\u03ba\u03bf\u03c2 \u03c4\u03b7\u03c2 \u03c5\u03c0\u03bf\u03c4\u03b5\u03af\u03bd\u03bf\u03c5\u03c3\u03b1\u03c2. (\u0395\u03b9\u03ba\u03cc\u03bd\u03b1 \u03b1\u03c0\u03cc\u00a0<a href=\"https:\/\/en.wikipedia.org\/wiki\/Sine_and_cosine#\/media\/File:Trigono_sine_en2.svg\" target=\"blank\" rel=\"noopener\">Dnu72<\/a>\u00a0\u2013\u00a0<a href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/3.0\" target=\"blank\" rel=\"noopener\">CC BY-SA 3.0.<\/a>\u00a0)<\/p>\n<\/div>\n<p>\u039f\u03b9 \u03c3\u03c5\u03bd\u03b1\u03c1\u03c4\u03ae\u03c3\u03b5\u03b9\u03c2 \u03b7\u03bc\u03b9\u03c4\u03cc\u03bd\u03bf\u03c5 \u03ba\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b7\u03bc\u03b9\u03c4\u03cc\u03bd\u03bf\u03c5 \u03bc\u03c0\u03bf\u03c1\u03bf\u03cd\u03bd \u03bd\u03b1 \u03ba\u03ac\u03bd\u03bf\u03c5\u03bd \u03c0\u03bf\u03bb\u03bb\u03ac \u03c0\u03b5\u03c1\u03b9\u03c3\u03c3\u03cc\u03c4\u03b5\u03c1\u03b1 \u03b1\u03c0\u03cc \u03c4\u03bf \u03bd\u03b1 \u03c3\u03b1\u03c2 \u03b2\u03bf\u03b7\u03b8\u03ae\u03c3\u03bf\u03c5\u03bd \u03bd\u03b1 \u03bb\u03cd\u03c3\u03b5\u03c4\u03b5 \u03c0\u03c1\u03bf\u03b2\u03bb\u03ae\u03bc\u03b1\u03c4\u03b1 \u03b3\u03b5\u03c9\u03bc\u03b5\u03c4\u03c1\u03af\u03b1\u03c2. \u039c\u03c0\u03bf\u03c1\u03bf\u03cd\u03bd \u03bd\u03b1 \u03c7\u03c1\u03b7\u03c3\u03b9\u03bc\u03bf\u03c0\u03bf\u03b9\u03b7\u03b8\u03bf\u03cd\u03bd \u03b3\u03b9\u03b1 \u03c4\u03b7 \u03b4\u03b7\u03bc\u03b9\u03bf\u03c5\u03c1\u03b3\u03af\u03b1 \u03bf\u03c0\u03bf\u03b9\u03b1\u03c3\u03b4\u03ae\u03c0\u03bf\u03c4\u03b5 \u03ba\u03c5\u03bc\u03b1\u03c4\u03bf\u03bc\u03bf\u03c1\u03c6\u03ae\u03c2 &#8211; \u03c4\u03b7 \u03bc\u03bf\u03c5\u03c3\u03b9\u03ba\u03ae \u03c0\u03bf\u03c5 \u03b1\u03ba\u03bf\u03cd\u03c4\u03b5, \u03c4\u03bf \u03c8\u03b7\u03c6\u03b9\u03b1\u03ba\u03cc \u03c3\u03ae\u03bc\u03b1 \u03c0\u03bf\u03c5 \u03c3\u03c4\u03ad\u03bb\u03bd\u03b5\u03c4\u03b5 \u03bc\u03ad\u03c3\u03c9 wifi, \u03b1\u03ba\u03cc\u03bc\u03b1 \u03ba\u03b1\u03b9 \u03c4\u03bf \u03ba\u03cd\u03bc\u03b1 \u03c3\u03c4\u03b7 \u03b8\u03ac\u03bb\u03b1\u03c3\u03c3\u03b1 &#8211; \u03b1\u03bd\u03b5\u03be\u03ac\u03c1\u03c4\u03b7\u03c4\u03b1 \u03b1\u03c0\u03cc \u03c4\u03bf \u03c0\u03cc\u03c3\u03bf \u03c0\u03b5\u03c1\u03af\u03c0\u03bb\u03bf\u03ba\u03b5\u03c2 \u03bc\u03c0\u03bf\u03c1\u03b5\u03af \u03bd\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b1\u03c5\u03c4\u03ad\u03c2 \u03bf\u03b9 \u03c6\u03c5\u03c3\u03b9\u03ba\u03ad\u03c2 \u03ae \u03b1\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03b3\u03b5\u03bd\u03b5\u03af\u03c2 \u03c4\u03b1\u03bb\u03b1\u03bd\u03c4\u03ce\u03c3\u03b5\u03b9\u03c2.<\/p>\n<h3>\u0393\u03cd\u03c1\u03bd\u03b1 \u03ba\u03b1\u03b9 \u03b3\u03cd\u03c1\u03bf<\/h3>\n<p>\u03a4\u03bf \u03c0\u03c1\u03ce\u03c4\u03bf \u03b2\u03ae\u03bc\u03b1 \u03b3\u03b9\u03b1 \u03c4\u03b7\u03bd \u03ba\u03b1\u03c4\u03b1\u03c3\u03ba\u03b5\u03c5\u03ae \u03b5\u03bd\u03cc\u03c2 \u03ba\u03cd\u03bc\u03b1\u03c4\u03bf\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bd\u03b1 \u03c6\u03b1\u03bd\u03c4\u03b1\u03c3\u03c4\u03bf\u03cd\u03bc\u03b5 \u03ad\u03bd\u03b1\u03bd \u03ba\u03cd\u03ba\u03bb\u03bf \u03b1\u03ba\u03c4\u03af\u03bd\u03b1\u03c2 1 \u03c0\u03bf\u03c5 \u03c3\u03c7\u03b5\u03b4\u03b9\u03ac\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c3\u03b5\u00a0<a href=\"https:\/\/plus.maths.org\/content\/maths-minute-cartesian-coordinates\" target=\"blank\" rel=\"noopener\">\u03ba\u03b1\u03c1\u03c4\u03b5\u03c3\u03b9\u03b1\u03bd\u03ad\u03c2 \u03c3\u03c5\u03bd\u03c4\u03b5\u03c4\u03b1\u03b3\u03bc\u03ad\u03bd\u03b5\u03c2<\/a>\u00a0, \u03bc\u03b5 \u03c4\u03bf \u03ba\u03ad\u03bd\u03c4\u03c1\u03bf \u03c4\u03bf\u03c5 \u03ba\u03cd\u03ba\u03bb\u03bf\u03c5 \u03bd\u03b1 \u03b2\u03c1\u03af\u03c3\u03ba\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/6d59ce379172546a494adc769d75acee\/images\/img-0001.png?w=700&#038;ssl=1\" alt=\"$(x,y)=(0,0).$\" data-recalc-dims=\"1\" \/>\u03a6\u03b1\u03bd\u03c4\u03b1\u03c3\u03c4\u03b5\u03af\u03c4\u03b5 \u03cc\u03c4\u03b9 \u03ba\u03b9\u03bd\u03b5\u03af\u03c3\u03c4\u03b5 \u03b3\u03cd\u03c1\u03c9 \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03ba\u03cd\u03ba\u03bb\u03bf \u03b1\u03c1\u03b9\u03c3\u03c4\u03b5\u03c1\u03cc\u03c3\u03c4\u03c1\u03bf\u03c6\u03b1, \u03be\u03b5\u03ba\u03b9\u03bd\u03ce\u03bd\u03c4\u03b1\u03c2 \u03b1\u03c0\u03cc \u03c4\u03bf \u03b4\u03b5\u03be\u03b9\u03cc\u03c4\u03b5\u03c1\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf,\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/6d59ce379172546a494adc769d75acee\/images\/img-0002.png?w=700&#038;ssl=1\" alt=\"$(1,0)$\" data-recalc-dims=\"1\" \/>. \u0391\u03c6\u03bf\u03cd \u03ad\u03c7\u03b5\u03c4\u03b5 \u03c0\u03b5\u03c1\u03b9\u03c3\u03c4\u03c1\u03b1\u03c6\u03b5\u03af \u03c3\u03b5 \u03b3\u03c9\u03bd\u03af\u03b1\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/6d59ce379172546a494adc769d75acee\/images\/img-0003.png?w=700&#038;ssl=1\" alt=\"$\\\u03ac\u03bb\u03c6\u03b1 $\" data-recalc-dims=\"1\" \/>\u03bc\u03b9\u03ba\u03c1\u03cc\u03c4\u03b5\u03c1\u03b7 \u03b1\u03c0\u03cc 90 \u03bc\u03bf\u03af\u03c1\u03b5\u03c2 (\u03c0\u03bf\u03c5 \u03b1\u03bd\u03c4\u03b9\u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u03af \u03c3\u03b5 \u03bc\u03b9\u03ba\u03c1\u03cc\u03c4\u03b5\u03c1\u03b7 \u03b1\u03c0\u03cc \u03cc ,\u03c4\u03b9\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/6d59ce379172546a494adc769d75acee\/images\/img-0004.png?w=700&#038;ssl=1\" alt=\"$\\pi \/2$\" data-recalc-dims=\"1\" \/>\u03c3\u03b5 \u03b1\u03ba\u03c4\u03af\u03bd\u03b9\u03b1), \u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03c3\u03c4\u03bf \u03bf\u03c0\u03bf\u03af\u03bf\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/6d59ce379172546a494adc769d75acee\/images\/img-0005.png?w=700&#038;ssl=1\" alt=\"$(x,y)$\" data-recalc-dims=\"1\" \/>\u03b2\u03c1\u03af\u03c3\u03ba\u03b5\u03c3\u03c4\u03b5 \u03bf\u03c1\u03af\u03b6\u03b5\u03b9 \u03ad\u03bd\u03b1 \u03bf\u03c1\u03b8\u03bf\u03b3\u03ce\u03bd\u03b9\u03bf \u03c4\u03c1\u03af\u03b3\u03c9\u03bd\u03bf, \u03bc\u03b5 \u03b3\u03c9\u03bd\u03af\u03b5\u03c2\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/6d59ce379172546a494adc769d75acee\/images\/img-0006.png?w=700&#038;ssl=1\" alt=\"$(0,0), (x,0)$\" data-recalc-dims=\"1\" \/>\u03ba\u03b1\u03b9\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/6d59ce379172546a494adc769d75acee\/images\/img-0007.png?w=700&#038;ssl=1\" alt=\"$(x,y).$\" data-recalc-dims=\"1\" \/>\u0397 \u03c5\u03c0\u03bf\u03c4\u03b5\u03af\u03bd\u03bf\u03c5\u03c3\u03b1 \u03b1\u03c5\u03c4\u03bf\u03cd \u03c4\u03bf\u03c5 \u03c4\u03c1\u03b9\u03b3\u03ce\u03bd\u03bf\u03c5 \u03ad\u03c7\u03b5\u03b9 \u03bc\u03ae\u03ba\u03bf\u03c2 1 \u03b5\u03c0\u03b5\u03b9\u03b4\u03ae \u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/6d59ce379172546a494adc769d75acee\/images\/img-0005.png?w=700&#038;ssl=1\" alt=\"$(x,y)$\" data-recalc-dims=\"1\" \/>\u03b2\u03c1\u03af\u03c3\u03ba\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf \u039f \u03ba\u03cd\u03ba\u03bb\u03bf\u03c2 \u03c4\u03b7\u03c2 \u03b1\u03ba\u03c4\u03af\u03bd\u03b1\u03c2 \u03bc\u03b1\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 1. \u0393\u03b9\u03b1 \u03b1\u03c5\u03c4\u03cc \u03c4\u03bf \u03c4\u03c1\u03af\u03b3\u03c9\u03bd\u03bf \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/6d59ce379172546a494adc769d75acee\/images\/img-0008.png?w=700&#038;ssl=1\" alt=\"$cos(\\alpha )=x$\" data-recalc-dims=\"1\" \/>\u03ba\u03b1\u03b9<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/6d59ce379172546a494adc769d75acee\/images\/img-0009.png?w=700&#038;ssl=1\" alt=\"$sin(\\alpha )=y.$\" data-recalc-dims=\"1\" \/><\/p>\n<div class=\"centreimage\">\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/content\/sites\/plus.maths.org\/files\/articles\/2022\/HYD2\/unit_circle.png?resize=400%2C400&#038;ssl=1\" alt=\"\" width=\"400\" height=\"400\" data-recalc-dims=\"1\" \/><\/p>\n<p>\u0388\u03bd\u03b1 \u03bf\u03c1\u03b8\u03bf\u03b3\u03ce\u03bd\u03b9\u03bf \u03c4\u03c1\u03af\u03b3\u03c9\u03bd\u03bf \u03c0\u03bf\u03c5 \u03c3\u03c7\u03b7\u03bc\u03b1\u03c4\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03b1\u03c0\u03cc \u03ad\u03bd\u03b1 \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03c4\u03bf\u03c5 \u03bc\u03bf\u03bd\u03b1\u03b4\u03b9\u03b1\u03af\u03bf\u03c5 \u03ba\u03cd\u03ba\u03bb\u03bf\u03c5.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>\u039c\u03c0\u03bf\u03c1\u03b5\u03af\u03c4\u03b5 \u03bd\u03b1 \u03c3\u03c5\u03bd\u03b5\u03c7\u03af\u03c3\u03b5\u03c4\u03b5 \u03bd\u03b1 \u03ba\u03b9\u03bd\u03b5\u03af\u03c3\u03c4\u03b5 \u03b3\u03cd\u03c1\u03c9 \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03ba\u03cd\u03ba\u03bb\u03bf \u03b1\u03c1\u03b9\u03c3\u03c4\u03b5\u03c1\u03cc\u03c3\u03c4\u03c1\u03bf\u03c6\u03b1 \u03b3\u03b9\u03b1 \u03bd\u03b1 \u03ba\u03ac\u03bd\u03b5\u03c4\u03b5 \u03c4\u03b7 \u03b3\u03c9\u03bd\u03af\u03b1\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/eb329facaafea0b5e849e53f3ac1aa89\/images\/img-0001.png?w=700&#038;ssl=1\" alt=\"$\\\u03ac\u03bb\u03c6\u03b1 $\" data-recalc-dims=\"1\" \/>\u03bc\u03b5\u03b3\u03b1\u03bb\u03cd\u03c4\u03b5\u03c1\u03b7 \u03b1\u03c0\u03cc\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/eb329facaafea0b5e849e53f3ac1aa89\/images\/img-0002.png?w=700&#038;ssl=1\" alt=\"$\\pi \/2.$\" data-recalc-dims=\"1\" \/>\u038c\u03c4\u03b1\u03bd \u03c4\u03bf \u03ba\u03ac\u03bd\u03b5\u03c4\u03b5 \u03b1\u03c5\u03c4\u03cc, \u03c4\u03bf \u03c4\u03c1\u03af\u03b3\u03c9\u03bd\u03bf \u03bc\u03b5 \u03b3\u03c9\u03bd\u03af\u03b5\u03c2\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/eb329facaafea0b5e849e53f3ac1aa89\/images\/img-0003.png?w=700&#038;ssl=1\" alt=\"$(0,0), (x,0)$\" data-recalc-dims=\"1\" \/>\u03ba\u03b1\u03b9\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/eb329facaafea0b5e849e53f3ac1aa89\/images\/img-0004.png?w=700&#038;ssl=1\" alt=\"$(x,y)$\" data-recalc-dims=\"1\" \/>\u03b4\u03b5\u03bd \u03ad\u03c7\u03b5\u03b9 \u03c0\u03bb\u03ad\u03bf\u03bd\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/eb329facaafea0b5e849e53f3ac1aa89\/images\/img-0001.png?w=700&#038;ssl=1\" alt=\"$\\\u03ac\u03bb\u03c6\u03b1 $\" data-recalc-dims=\"1\" \/>\u03c9\u03c2 \u03bc\u03af\u03b1 \u03b1\u03c0\u03cc \u03c4\u03b9\u03c2 \u03b3\u03c9\u03bd\u03af\u03b5\u03c2 \u03c4\u03bf\u03c5.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"centreimage\">\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/content\/sites\/plus.maths.org\/files\/articles\/2022\/HYD2\/unit_circle3.png?resize=383%2C388&#038;ssl=1\" alt=\"\" width=\"383\" height=\"388\" data-recalc-dims=\"1\" \/>\u038c\u03c4\u03b1\u03bd \u03c4\u03bf \u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bc\u03b5\u03b3\u03b1\u03bb\u03cd\u03c4\u03b5\u03c1\u03bf \u03b1\u03c0\u03cc \u03c0\/2, \u03c4\u03cc\u03c4\u03b5 \u03c4\u03bf \u03bf\u03c1\u03b8\u03bf\u03b3\u03ce\u03bd\u03b9\u03bf \u03c4\u03c1\u03af\u03b3\u03c9\u03bd\u03bf \u03c0\u03bf\u03c5 \u03c3\u03c7\u03b7\u03bc\u03b1\u03c4\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03b1\u03c0\u03cc \u03ad\u03bd\u03b1 \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03c4\u03bf\u03c5 \u03bc\u03bf\u03bd\u03b1\u03b4\u03b9\u03b1\u03af\u03bf\u03c5 \u03ba\u03cd\u03ba\u03bb\u03bf\u03c5 \u03b4\u03b5\u03bd \u03c0\u03b5\u03c1\u03b9\u03ad\u03c7\u03b5\u03b9 \u03c0\u03bb\u03ad\u03bf\u03bd \u03c4\u03b7 \u03b3\u03c9\u03bd\u03af\u03b1 \u03b1.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>\u03a9\u03c3\u03c4\u03cc\u03c3\u03bf, \u03b4\u03b5\u03bd \u03c5\u03c0\u03ac\u03c1\u03c7\u03b5\u03b9 \u03c4\u03af\u03c0\u03bf\u03c4\u03b1 \u03c0\u03bf\u03c5 \u03bd\u03b1 \u03c3\u03b1\u03c2 \u03b5\u03bc\u03c0\u03bf\u03b4\u03af\u03b6\u03b5\u03b9 \u03bd\u03b1 \u03b5\u03c0\u03b5\u03ba\u03c4\u03b5\u03af\u03bd\u03b5\u03c4\u03b5 \u03c4\u03bf\u03c5\u03c2 \u03bf\u03c1\u03b9\u03c3\u03bc\u03bf\u03cd\u03c2 \u03c4\u03bf\u03c5 \u03b7\u03bc\u03b9\u03c4\u03cc\u03bd\u03bf\u03c5 \u03ba\u03b1\u03b9 \u03c4\u03bf\u03c5 \u03c3\u03c5\u03bd\u03b7\u03bc\u03af\u03c4\u03bf\u03bd\u03bf\u03c2\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/f2e0b291c35aa40f7a394d5b8bdb24bd\/images\/img-0001.png?w=700&#038;ssl=1\" alt=\"$\\\u03ac\u03bb\u03c6\u03b1 $\" data-recalc-dims=\"1\" \/>\u03c3\u03b5 \u03b1\u03bd\u03b1\u03bb\u03bf\u03b3\u03af\u03b1 \u03bc\u03b5 \u03b1\u03c5\u03c4\u03cc \u03c0\u03bf\u03c5 \u03b5\u03af\u03c7\u03b1\u03bc\u03b5 \u03c0\u03c1\u03b9\u03bd:<\/p>\n<table id=\"a0000000002\" class=\"equation\" width=\"100%\" cellspacing=\"0\" cellpadding=\"7\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/f2e0b291c35aa40f7a394d5b8bdb24bd\/images\/img-0002.png?w=700&#038;ssl=1\" alt=\"\\[ \\cos {(\\alpha )}=x\\; \\; \\; \\; \\mbox{and}\\; \\; \\; \\sin {(\\alpha )}=y, \\]\" data-recalc-dims=\"1\" \/><\/td>\n<td><\/td>\n<td class=\"eqnnum\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u03c0\u03bf\u03cd\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/f2e0b291c35aa40f7a394d5b8bdb24bd\/images\/img-0003.png?w=700&#038;ssl=1\" alt=\"$(x,y)$\" data-recalc-dims=\"1\" \/>\u03b5\u03af\u03bd\u03b1\u03b9 \u03bf\u03b9 \u03c3\u03c5\u03bd\u03c4\u03b5\u03c4\u03b1\u03b3\u03bc\u03ad\u03bd\u03b5\u03c2 \u03c4\u03bf\u03c5 \u03c3\u03b7\u03bc\u03b5\u03af\u03bf\u03c5 \u03c3\u03c4\u03bf \u03bf\u03c0\u03bf\u03af\u03bf \u03b2\u03c1\u03af\u03c3\u03ba\u03b5\u03c3\u03c4\u03b5.<\/p>\n<p>&nbsp;<\/p>\n<p>\u03a4\u03b9 \u03c3\u03c5\u03bc\u03b2\u03b1\u03af\u03bd\u03b5\u03b9 \u03bc\u03b5 \u03c4\u03bf \u03b7\u03bc\u03af\u03c4\u03bf\u03bd\u03bf \u03ba\u03b1\u03b9 \u03c4\u03bf \u03c3\u03c5\u03bd\u03b7\u03bc\u03af\u03c4\u03bf\u03bd\u03bf \u03ba\u03b1\u03b8\u03ce\u03c2 \u03ba\u03b9\u03bd\u03b5\u03af\u03c3\u03c4\u03b5 \u03bc\u03b9\u03b1 \u03c6\u03bf\u03c1\u03ac \u03b3\u03cd\u03c1\u03c9 \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03ba\u03cd\u03ba\u03bb\u03bf; \u03a3\u03b5 \u03ad\u03bd\u03b1 \u03ba\u03cd\u03ba\u03bb\u03c9\u03bc\u03b1 \u03c4\u03bf\u03c5 \u03ba\u03cd\u03ba\u03bb\u03bf\u03c5 \u03b8\u03b1 \u03ad\u03c7\u03b5\u03c4\u03b5 \u03b3\u03c5\u03c1\u03af\u03c3\u03b5\u03b9 \u03ba\u03b1\u03c4\u03ac \u03b3\u03c9\u03bd\u03af\u03b1\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/11c0a1ed8e414d3421ed52ec68b26de7\/images\/img-0001.png?w=700&#038;ssl=1\" alt=\"$2\\pi $\" data-recalc-dims=\"1\" \/>. \u039a\u03b1\u03b8\u03ce\u03c2 \u03ba\u03b9\u03bd\u03b7\u03b8\u03ae\u03ba\u03b1\u03c4\u03b5 \u03b3\u03cd\u03c1\u03c9 \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03ba\u03cd\u03ba\u03bb\u03bf \u03b7 \u03ba\u03b1\u03c4\u03b1\u03ba\u03cc\u03c1\u03c5\u03c6\u03b7 \u03c3\u03c5\u03bd\u03c4\u03b5\u03c4\u03b1\u03b3\u03bc\u03ad\u03bd\u03b7 \u03c3\u03b1\u03c2 (\u03c4\u03bf \u03b7\u03bc\u03af\u03c4\u03bf\u03bd\u03bf) \u03be\u03b5\u03ba\u03b9\u03bd\u03bf\u03cd\u03c3\u03b5 \u03b1\u03c0\u03cc\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/11c0a1ed8e414d3421ed52ec68b26de7\/images\/img-0002.png?w=700&#038;ssl=1\" alt=\"$0 $\" data-recalc-dims=\"1\" \/>\u03ba\u03b1\u03b9 \u03b1\u03c5\u03be\u03b1\u03bd\u03cc\u03c4\u03b1\u03bd \u03c3\u03c4\u03b1\u03b8\u03b5\u03c1\u03ac \u03bc\u03ad\u03c7\u03c1\u03b9 \u03bd\u03b1 \u03c6\u03c4\u03ac\u03c3\u03b5\u03b9 \u03c3\u03c4\u03bf \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03bf\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/11c0a1ed8e414d3421ed52ec68b26de7\/images\/img-0003.png?w=700&#038;ssl=1\" alt=\"$1$\" data-recalc-dims=\"1\" \/>\u03cc\u03c4\u03b1\u03bd \u03b2\u03c1\u03b9\u03c3\u03ba\u03cc\u03c3\u03b1\u03c3\u03c4\u03b1\u03bd \u03c3\u03c4\u03b7\u03bd \u03ba\u03bf\u03c1\u03c5\u03c6\u03ae \u03c4\u03bf\u03c5 \u03ba\u03cd\u03ba\u03bb\u03bf\u03c5. \u039a\u03b1\u03b8\u03ce\u03c2 \u03c3\u03c5\u03bd\u03b5\u03c7\u03af\u03b6\u03b1\u03c4\u03b5 \u03bd\u03b1 \u03c4\u03bf \u03bc\u03b5\u03c4\u03b1\u03ba\u03b9\u03bd\u03b5\u03af\u03c4\u03b5, \u03ad\u03c0\u03b5\u03c6\u03c4\u03b5\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/11c0a1ed8e414d3421ed52ec68b26de7\/images\/img-0002.png?w=700&#038;ssl=1\" alt=\"$0 $\" data-recalc-dims=\"1\" \/>\u03be\u03b1\u03bd\u03ac \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ba\u03ac\u03c4\u03c9 \u03bc\u03b5 \u03c3\u03c5\u03bc\u03bc\u03b5\u03c4\u03c1\u03b9\u03ba\u03cc \u03c4\u03c1\u03cc\u03c0\u03bf, \u03c0\u03c1\u03b9\u03bd \u03c6\u03c4\u03ac\u03c3\u03b5\u03c4\u03b5 \u03c3\u03c4\u03bf \u03b5\u03bb\u03ac\u03c7\u03b9\u03c3\u03c4\u03bf\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/11c0a1ed8e414d3421ed52ec68b26de7\/images\/img-0004.png?w=700&#038;ssl=1\" alt=\"-1$\" data-recalc-dims=\"1\" \/>, \u03ba\u03b1\u03b9 \u03c4\u03b5\u03bb\u03b9\u03ba\u03ac \u03b1\u03bd\u03b5\u03b2\u03b5\u03af\u03c4\u03b5 \u03be\u03b1\u03bd\u03ac \u03c3\u03c4\u03bf 0.<\/p>\n<p>\u0395\u03ac\u03bd \u03c3\u03c7\u03b5\u03b4\u03b9\u03ac\u03c3\u03b5\u03c4\u03b5 \u03c0\u03ce\u03c2 \u03b7 \u03ba\u03b1\u03c4\u03b1\u03ba\u03cc\u03c1\u03c5\u03c6\u03b7 \u03c3\u03c5\u03bd\u03c4\u03b5\u03c4\u03b1\u03b3\u03bc\u03ad\u03bd\u03b7 (\u03bc\u03b5 \u03ba\u03cc\u03ba\u03ba\u03b9\u03bd\u03bf \u03c3\u03c4\u03bf \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c3\u03c7\u03ae\u03bc\u03b1) \u03bc\u03b5\u03c4\u03b1\u03b2\u03ac\u03bb\u03bb\u03b5\u03c4\u03b1\u03b9 \u03bc\u03b5 \u03c4\u03b7 \u03b3\u03c9\u03bd\u03af\u03b1 \u03c0\u03bf\u03c5 \u03c3\u03c4\u03c1\u03ad\u03c6\u03b5\u03c4\u03b1\u03b9 (\u03b1\u03c0\u03cc 0 \u03c3\u03b5\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/2b28ca241db87e6248f9a8683336f0f0\/images\/img-0001.png?w=700&#038;ssl=1\" alt=\"$2\\pi $\" data-recalc-dims=\"1\" \/>) \u03b8\u03b1 \u03ad\u03c7\u03b5\u03c4\u03b5 \u03ad\u03bd\u03b1 \u03ba\u03b1\u03bd\u03bf\u03bd\u03b9\u03ba\u03cc \u03c3\u03c7\u03ae\u03bc\u03b1 \u03ba\u03cd\u03bc\u03b1\u03c4\u03bf\u03c2. \u039e\u03b5\u03ba\u03b9\u03bd\u03ac \u03b1\u03c0\u03cc \u03c4\u03bf 0, \u03b1\u03bd\u03b5\u03b2\u03b1\u03af\u03bd\u03b5\u03b9 \u03c3\u03c4\u03bf \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03bf 1, \u03bc\u03b5\u03c4\u03ac \u03bc\u03b5\u03b9\u03ce\u03bd\u03b5\u03c4\u03b1\u03b9 \u03be\u03b1\u03bd\u03ac \u03c3\u03c4\u03bf 0, \u03c0\u03c1\u03b9\u03bd \u03c0\u03ad\u03c3\u03b5\u03b9 \u03c3\u03c4\u03bf \u03b5\u03bb\u03ac\u03c7\u03b9\u03c3\u03c4\u03bf \u03c4\u03bf\u03c5 -1 \u03ba\u03b1\u03b9 \u03be\u03b1\u03bd\u03ac \u03c3\u03c4\u03bf 0.<\/p>\n<div class=\"centreimage\">\n<p><img decoding=\"async\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/content\/sites\/plus.maths.org\/files\/articles\/2022\/HYD2\/waves.png?w=700&#038;ssl=1\" data-recalc-dims=\"1\" \/>\u03a4\u03bf \u03ba\u03cc\u03ba\u03ba\u03b9\u03bd\u03bf \u03ba\u03cd\u03bc\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03bf \u03b7\u03bc\u03af\u03c4\u03bf\u03bd\u03bf \u03c4\u03b7\u03c2 \u03b3\u03c9\u03bd\u03af\u03b1\u03c2 \u03c0\u03bf\u03c5 \u03b1\u03c0\u03b5\u03b9\u03ba\u03bf\u03bd\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03ad\u03bd\u03b1\u03bd\u03c4\u03b9 \u03c4\u03b7\u03c2 \u03b3\u03c9\u03bd\u03af\u03b1\u03c2 (\u03c0\u03bf\u03c5 \u03c0\u03c1\u03bf\u03ad\u03c1\u03c7\u03b5\u03c4\u03b1\u03b9 \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd \u03ba\u03b1\u03c4\u03b1\u03ba\u03cc\u03c1\u03c5\u03c6\u03b7 \u03c3\u03c5\u03bd\u03c4\u03b5\u03c4\u03b1\u03b3\u03bc\u03ad\u03bd\u03b7) \u03ba\u03b1\u03b9 \u03c4\u03bf \u03bc\u03c0\u03bb\u03b5 \u03ba\u03cd\u03bc\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03bf \u03c3\u03c5\u03bd\u03b7\u03bc\u03af\u03c4\u03bf\u03bd\u03bf \u03c4\u03b7\u03c2 \u03b3\u03c9\u03bd\u03af\u03b1\u03c2 \u03c0\u03bf\u03c5 \u03b1\u03c0\u03b5\u03b9\u03ba\u03bf\u03bd\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03ad\u03bd\u03b1\u03bd\u03c4\u03b9 \u03c4\u03b7\u03c2 \u03b3\u03c9\u03bd\u03af\u03b1\u03c2 (\u03c0\u03bf\u03c5 \u03c0\u03c1\u03bf\u03ad\u03c1\u03c7\u03b5\u03c4\u03b1\u03b9 \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd \u03bf\u03c1\u03b9\u03b6\u03cc\u03bd\u03c4\u03b9\u03b1 \u03c3\u03c5\u03bd\u03c4\u03b5\u03c4\u03b1\u03b3\u03bc\u03ad\u03bd\u03b7).<\/p>\n<\/div>\n<p>\u0391\u03c5\u03c4\u03cc \u03c4\u03bf \u03bc\u03bf\u03c4\u03af\u03b2\u03bf \u03ba\u03c5\u03bc\u03ac\u03c4\u03c9\u03bd \u03b5\u03c0\u03b1\u03bd\u03b1\u03bb\u03b1\u03bc\u03b2\u03ac\u03bd\u03b5\u03c4\u03b1\u03b9 \u03ba\u03b1\u03b8\u03ce\u03c2 \u03c3\u03c5\u03bd\u03b5\u03c7\u03af\u03b6\u03b5\u03c4\u03b5 \u03bd\u03b1 \u03c0\u03b5\u03c1\u03b9\u03c6\u03ad\u03c1\u03b5\u03c3\u03c4\u03b5 \u03b3\u03b9\u03b1 \u03b4\u03b5\u03cd\u03c4\u03b5\u03c1\u03b7 \u03c6\u03bf\u03c1\u03ac \u03b3\u03cd\u03c1\u03c9 \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03ba\u03cd\u03ba\u03bb\u03bf \u03c3\u03b1\u03c2, \u03b1\u03c5\u03be\u03ac\u03bd\u03bf\u03bd\u03c4\u03b1\u03c2 \u03c4\u03b7 \u03b3\u03c9\u03bd\u03af\u03b1 \u03c0\u03bf\u03c5 \u03c3\u03c4\u03c1\u03af\u03c8\u03b1\u03c4\u03b5 \u03b1\u03c0\u03cc\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/85f16ad980a088ba4ad239ed22822739\/images\/img-0001.png?w=700&#038;ssl=1\" alt=\"$2\\pi $\" data-recalc-dims=\"1\" \/>\u03ad\u03c9\u03c2\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/85f16ad980a088ba4ad239ed22822739\/images\/img-0002.png?w=700&#038;ssl=1\" alt=\"$4\\pi , $\" data-recalc-dims=\"1\" \/>\u03ba\u03b1\u03b9 \u03b3\u03b9\u03b1 \u03c4\u03c1\u03af\u03c4\u03b7 \u03c6\u03bf\u03c1\u03ac \u03c4\u03b1\u03be\u03b9\u03b4\u03b5\u03cd\u03bf\u03bd\u03c4\u03b1\u03c2 \u03b1\u03c0\u03cc\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/85f16ad980a088ba4ad239ed22822739\/images\/img-0003.png?w=700&#038;ssl=1\" alt=\"$4\\pi $\" data-recalc-dims=\"1\" \/>\u03ad\u03c9\u03c2\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/85f16ad980a088ba4ad239ed22822739\/images\/img-0004.png?w=700&#038;ssl=1\" alt=\"$6\\pi $\" data-recalc-dims=\"1\" \/>\u03ba\u03b1\u03b9 \u03bf\u03cd\u03c4\u03c9 \u03ba\u03b1\u03b8\u03b5\u03be\u03ae\u03c2. \u039a\u03b1\u03c4\u03b1\u03bb\u03ae\u03b3\u03b5\u03b9\u03c2 \u03bc\u03b5 \u03ad\u03bd\u03b1 \u03b1\u03c0\u03b5\u03af\u03c1\u03c9\u03c2 \u03bc\u03b1\u03ba\u03c1\u03cd, \u03b1\u03c0\u03cc\u03bb\u03c5\u03c4\u03b1 \u03ba\u03b1\u03bd\u03bf\u03bd\u03b9\u03ba\u03cc \u03ba\u03cd\u03bc\u03b1. \u039f\u03b9 \u03bf\u03c1\u03b9\u03b6\u03cc\u03bd\u03c4\u03b9\u03b5\u03c2 \u03c3\u03c5\u03bd\u03c4\u03b5\u03c4\u03b1\u03b3\u03bc\u03ad\u03bd\u03b5\u03c2 (\u03c4\u03bf \u03c3\u03c5\u03bd\u03b7\u03bc\u03af\u03c4\u03bf\u03bd\u03bf) \u03b4\u03af\u03bd\u03bf\u03c5\u03bd \u03ad\u03bd\u03b1 \u03c0\u03b1\u03c1\u03cc\u03bc\u03bf\u03b9\u03bf \u03c4\u03ad\u03bb\u03b5\u03b9\u03bf \u03ba\u03cd\u03bc\u03b1 (\u03c7\u03b1\u03c1\u03b1\u03ba\u03c4\u03b7\u03c1\u03b9\u03c3\u03bc\u03ad\u03bd\u03bf \u03bc\u03b5 \u03bc\u03c0\u03bb\u03b5 \u03c7\u03c1\u03ce\u03bc\u03b1 \u03b5\u03c0\u03ac\u03bd\u03c9), \u03bc\u03b5\u03c4\u03b1\u03c4\u03bf\u03c0\u03b9\u03c3\u03bc\u03ad\u03bd\u03bf \u03ad\u03bd\u03b1\u03bd\u03c4\u03b9 \u03c4\u03bf\u03c5 \u03c0\u03c1\u03ce\u03c4\u03bf\u03c5 \u03ba\u03b1\u03c4\u03ac \u03bc\u03b9\u03b1 \u03b1\u03c0\u03cc\u03c3\u03c4\u03b1\u03c3\u03b7\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/85f16ad980a088ba4ad239ed22822739\/images\/img-0005.png?w=700&#038;ssl=1\" alt=\"$\\pi \/2$\" data-recalc-dims=\"1\" \/>\u03ba\u03b1\u03c4\u03ac \u03bc\u03ae\u03ba\u03bf\u03c2 \u03c4\u03bf\u03c5 \u03bf\u03c1\u03b9\u03b6\u03cc\u03bd\u03c4\u03b9\u03bf\u03c5 \u03ac\u03be\u03bf\u03bd\u03b1. \u0391\u03c5\u03c4\u03cc \u03b5\u03c0\u03b5\u03ba\u03c4\u03b5\u03af\u03bd\u03b5\u03b9 \u03c4\u03bf\u03bd \u03bf\u03c1\u03b9\u03c3\u03bc\u03cc \u03bc\u03b1\u03c2 \u03b3\u03b9\u03b1 \u03c4\u03bf \u03b7\u03bc\u03af\u03c4\u03bf\u03bd\u03bf \u03ba\u03b1\u03b9 \u03c4\u03bf \u03c3\u03c5\u03bd\u03b7\u03bc\u03af\u03c4\u03bf\u03bd\u03bf \u03c3\u03b5 \u03b3\u03c9\u03bd\u03af\u03b5\u03c2 \u03bc\u03b5\u03b3\u03b1\u03bb\u03cd\u03c4\u03b5\u03c1\u03b5\u03c2 \u03b1\u03c0\u03cc\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/85f16ad980a088ba4ad239ed22822739\/images\/img-0005.png?w=700&#038;ssl=1\" alt=\"$\\pi \/2$\" data-recalc-dims=\"1\" \/>. (\u039c\u03c0\u03bf\u03c1\u03b5\u03af\u03c4\u03b5 \u03b5\u03c0\u03af\u03c3\u03b7\u03c2 \u03bd\u03b1 \u03bf\u03c1\u03af\u03c3\u03b5\u03c4\u03b5 \u03c4\u03bf \u03b7\u03bc\u03af\u03c4\u03bf\u03bd\u03bf \u03ba\u03b1\u03b9 \u03c4\u03bf \u03c3\u03c5\u03bd\u03b7\u03bc\u03af\u03c4\u03bf\u03bd\u03bf \u03b3\u03b9\u03b1 \u03c4\u03bf\u03c5\u03c2 \u03b1\u03c1\u03bd\u03b7\u03c4\u03b9\u03ba\u03bf\u03cd\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03cd\u03c2 \u03c0\u03b5\u03c1\u03b9\u03c3\u03c4\u03c1\u03ad\u03c6\u03bf\u03bd\u03c4\u03b1\u03c2 \u03c4\u03bf\u03bd \u03ba\u03cd\u03ba\u03bb\u03bf \u03ba\u03b1\u03c4\u03ac \u03c4\u03b7 \u03c6\u03bf\u03c1\u03ac \u03c4\u03c9\u03bd \u03b4\u03b5\u03b9\u03ba\u03c4\u03ce\u03bd \u03c4\u03bf\u03c5 \u03c1\u03bf\u03bb\u03bf\u03b3\u03b9\u03bf\u03cd.)<\/p>\n<div class=\"centreimage\">\n<p><img decoding=\"async\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/content\/sites\/plus.maths.org\/files\/articles\/2022\/HYD2\/waves2.png?w=700&#038;ssl=1\" data-recalc-dims=\"1\" \/>\u03a4\u03bf \u03ba\u03cc\u03ba\u03ba\u03b9\u03bd\u03bf \u03ba\u03cd\u03bc\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03bf \u03b7\u03bc\u03af\u03c4\u03bf\u03bd\u03bf \u03c4\u03b7\u03c2 \u03b3\u03c9\u03bd\u03af\u03b1\u03c2 \u03c0\u03bf\u03c5 \u03b1\u03c0\u03b5\u03b9\u03ba\u03bf\u03bd\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03ad\u03bd\u03b1\u03bd\u03c4\u03b9 \u03c4\u03b7\u03c2 \u03b3\u03c9\u03bd\u03af\u03b1\u03c2 (\u03c0\u03bf\u03c5 \u03c0\u03c1\u03bf\u03ad\u03c1\u03c7\u03b5\u03c4\u03b1\u03b9 \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd \u03ba\u03b1\u03c4\u03b1\u03ba\u03cc\u03c1\u03c5\u03c6\u03b7 \u03c3\u03c5\u03bd\u03c4\u03b5\u03c4\u03b1\u03b3\u03bc\u03ad\u03bd\u03b7) \u03ba\u03b1\u03b9 \u03c4\u03bf \u03bc\u03c0\u03bb\u03b5 \u03ba\u03cd\u03bc\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03bf \u03c3\u03c5\u03bd\u03b7\u03bc\u03af\u03c4\u03bf\u03bd\u03bf \u03c4\u03b7\u03c2 \u03b3\u03c9\u03bd\u03af\u03b1\u03c2 \u03c0\u03bf\u03c5 \u03b1\u03c0\u03b5\u03b9\u03ba\u03bf\u03bd\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03ad\u03bd\u03b1\u03bd\u03c4\u03b9 \u03c4\u03b7\u03c2 \u03b3\u03c9\u03bd\u03af\u03b1\u03c2 (\u03c0\u03bf\u03c5 \u03c0\u03c1\u03bf\u03ad\u03c1\u03c7\u03b5\u03c4\u03b1\u03b9 \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd \u03bf\u03c1\u03b9\u03b6\u03cc\u03bd\u03c4\u03b9\u03b1 \u03c3\u03c5\u03bd\u03c4\u03b5\u03c4\u03b1\u03b3\u03bc\u03ad\u03bd\u03b7).<\/p>\n<\/div>\n<h3>\u03a3\u03c6\u03af\u03be\u03b9\u03bc\u03bf \u03ba\u03b1\u03b9 \u03c4\u03ad\u03bd\u03c4\u03c9\u03bc\u03b1<\/h3>\n<p>&nbsp;<\/p>\n<p>\u0395\u03af\u03b4\u03b1\u03bc\u03b5 \u03c4\u03ce\u03c1\u03b1 \u03c0\u03ce\u03c2 \u03b7 \u03ba\u03af\u03bd\u03b7\u03c3\u03b7 \u03b3\u03cd\u03c1\u03c9 \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03bc\u03bf\u03bd\u03b1\u03b4\u03b9\u03b1\u03af\u03bf \u03ba\u03cd\u03ba\u03bb\u03bf \u03bc\u03c0\u03bf\u03c1\u03b5\u03af \u03bd\u03b1 \u03bc\u03b1\u03c2 \u03b4\u03ce\u03c3\u03b5\u03b9 \u03b4\u03cd\u03bf \u03c3\u03c5\u03bd\u03b1\u03c1\u03c4\u03ae\u03c3\u03b5\u03b9\u03c2, \u03c4\u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u03b7\u03bc\u03b9\u03c4\u03cc\u03bd\u03bf\u03c5 \u03ba\u03b1\u03b9 \u03c4\u03b7 \u03c3\u03c5\u03bd\u03b7\u03bc\u03af\u03c4\u03bf\u03bd\u03bf, \u03ba\u03b1\u03b8\u03b5\u03bc\u03af\u03b1 \u03b1\u03c0\u03cc \u03c4\u03b9\u03c2 \u03bf\u03c0\u03bf\u03af\u03b5\u03c2 \u03c3\u03c5\u03bd\u03bf\u03b4\u03b5\u03cd\u03b5\u03c4\u03b1\u03b9 \u03b1\u03c0\u03cc \u03ad\u03bd\u03b1 \u03b3\u03c1\u03ac\u03c6\u03b7\u03bc\u03b1 \u03c0\u03bf\u03c5 \u03c0\u03b5\u03c1\u03b9\u03b3\u03c1\u03ac\u03c6\u03b5\u03b9 \u03ad\u03bd\u03b1 \u03ba\u03b1\u03bd\u03bf\u03bd\u03b9\u03ba\u03cc \u03ba\u03cd\u03bc\u03b1. \u038c\u03c4\u03b1\u03bd \u03c3\u03c5\u03bd\u03b1\u03bd\u03c4\u03ac\u03c4\u03b5 \u03b1\u03c5\u03c4\u03ad\u03c2 \u03c4\u03b9\u03c2 \u03c3\u03c5\u03bd\u03b1\u03c1\u03c4\u03ae\u03c3\u03b5\u03b9\u03c2 \u03c3\u03b5 \u03b2\u03b9\u03b2\u03bb\u03af\u03b1, \u03b7 \u03bc\u03b5\u03c4\u03b1\u03b2\u03bb\u03b7\u03c4\u03ae \u03c3\u03c5\u03bd\u03ae\u03b8\u03c9\u03c2 \u03ba\u03b1\u03bb\u03b5\u03af\u03c4\u03b1\u03b9\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/c8cbad5ecffe0cce506a4d08914779d2\/images\/img-0001.png?w=700&#038;ssl=1\" alt=\"$x$\" data-recalc-dims=\"1\" \/>\u03b1\u03bd\u03c4\u03af \u03b3\u03b9\u03b1\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/c8cbad5ecffe0cce506a4d08914779d2\/images\/img-0002.png?w=700&#038;ssl=1\" alt=\"$\\\u03ac\u03bb\u03c6\u03b1 $\" data-recalc-dims=\"1\" \/>, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03b8\u03b1 \u03b4\u03b5\u03af\u03c4\u03b5 \u03ba\u03ac\u03c4\u03b9 \u03c3\u03b1\u03bd<\/p>\n<table id=\"a0000000002\" class=\"equation\" width=\"100%\" cellspacing=\"0\" cellpadding=\"7\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/c8cbad5ecffe0cce506a4d08914779d2\/images\/img-0003.png?w=700&#038;ssl=1\" alt=\"\\[ f(x)=\\sin {(x)}\\; \\; \\; \\; \\mbox{and}\\; \\; \\; g(x)=cos{(x)}. \\]\" data-recalc-dims=\"1\" \/><\/td>\n<td><\/td>\n<td class=\"eqnnum\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u0398\u03b1 \u03bc\u03b5\u03c4\u03b1\u03b2\u03bf\u03cd\u03bc\u03b5 \u03c3\u03b5 \u03b1\u03c5\u03c4\u03cc\u03bd \u03c4\u03bf\u03bd \u03c3\u03c5\u03bc\u03b2\u03bf\u03bb\u03b9\u03c3\u03bc\u03cc \u03c4\u03ce\u03c1\u03b1. \u0391\u03c5\u03c4\u03cc \u03c3\u03b7\u03bc\u03b1\u03af\u03bd\u03b5\u03b9 \u03cc\u03c4\u03b9\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/c8cbad5ecffe0cce506a4d08914779d2\/images\/img-0001.png?w=700&#038;ssl=1\" alt=\"$x$\" data-recalc-dims=\"1\" \/>\u03c4\u03ce\u03c1\u03b1 \u03b1\u03bd\u03c4\u03b9\u03c0\u03c1\u03bf\u03c3\u03c9\u03c0\u03b5\u03cd\u03b5\u03b9 \u03b1\u03c5\u03c4\u03cc \u03c0\u03bf\u03c5 \u03ae\u03c4\u03b1\u03bd \u03b7 \u03b3\u03c9\u03bd\u03af\u03b1 \u03bc\u03b1\u03c2\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/c8cbad5ecffe0cce506a4d08914779d2\/images\/img-0002.png?w=700&#038;ssl=1\" alt=\"$\\\u03ac\u03bb\u03c6\u03b1 $\" data-recalc-dims=\"1\" \/>\u03ba\u03b1\u03b9 \u03cc\u03c4\u03b9 \u03b4\u03b5\u03bd \u03b1\u03bd\u03c4\u03b9\u03c0\u03c1\u03bf\u03c3\u03c9\u03c0\u03b5\u03cd\u03b5\u03b9 \u03c0\u03bb\u03ad\u03bf\u03bd \u03bc\u03b9\u03b1 \u03c3\u03c5\u03bd\u03c4\u03b5\u03c4\u03b1\u03b3\u03bc\u03ad\u03bd\u03b7 \u03b5\u03bd\u03cc\u03c2 \u03c3\u03b7\u03bc\u03b5\u03af\u03bf\u03c5 \u03c3\u03c4\u03bf\u03bd \u03bc\u03bf\u03bd\u03b1\u03b4\u03b9\u03b1\u03af\u03bf \u03ba\u03cd\u03ba\u03bb\u03bf. \u0391\u03c5\u03c4\u03cc \u03b5\u03af\u03bd\u03b1\u03b9 \u03b4\u03c5\u03bd\u03b7\u03c4\u03b9\u03ba\u03ac \u03bb\u03af\u03b3\u03bf \u03bc\u03c0\u03b5\u03c1\u03b4\u03b5\u03bc\u03ad\u03bd\u03bf, \u03b1\u03bb\u03bb\u03ac \u03bc\u03b5\u03af\u03bd\u03b5\u03c4\u03b5 \u03b5\u03ba\u03b5\u03af \u03ba\u03b1\u03b9 \u03b8\u03b1 \u03c4\u03bf \u03c3\u03c5\u03bd\u03b7\u03b8\u03af\u03c3\u03b5\u03c4\u03b5.<\/p>\n<p>&nbsp;<\/p>\n<p>\u03a4\u03bf\u00a0<em>\u03bc\u03ae\u03ba\u03bf\u03c2 \u03ba\u03cd\u03bc\u03b1\u03c4\u03bf\u03c2<\/em>\u00a0\u03b5\u03af\u03bd\u03b1\u03b9 \u03b7 \u03b1\u03c0\u03cc\u03c3\u03c4\u03b1\u03c3\u03b7 \u03bc\u03b5\u03c4\u03b1\u03be\u03cd \u03b4\u03cd\u03bf \u03ba\u03bf\u03c1\u03c5\u03c6\u03ce\u03bd \u03b5\u03bd\u03cc\u03c2 \u03ba\u03cd\u03bc\u03b1\u03c4\u03bf\u03c2, \u03ba\u03b1\u03b9 \u03c3\u03c4\u03bf \u03c0\u03b1\u03c1\u03ac\u03b4\u03b5\u03b9\u03b3\u03bc\u03ac \u03bc\u03b1\u03c2 \u03bc\u03ad\u03c7\u03c1\u03b9 \u03c3\u03c4\u03b9\u03b3\u03bc\u03ae\u03c2 \u03b1\u03c5\u03c4\u03cc \u03b5\u03af\u03bd\u03b1\u03b9\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/f7cab61d3dd0998c89aeae012dc0cb54\/images\/img-0001.png?w=700&#038;ssl=1\" alt=\"$2\\pi $\" data-recalc-dims=\"1\" \/>. \u0391\u03c5\u03c4\u03cc \u03c3\u03c5\u03bc\u03b2\u03b1\u03af\u03bd\u03b5\u03b9 \u03b5\u03c0\u03b5\u03b9\u03b4\u03ae \u03b3\u03b9\u03b1 \u03bd\u03b1 \u03c0\u03b5\u03c1\u03ac\u03c3\u03bf\u03c5\u03bc\u03b5 \u03b1\u03c0\u03cc \u03ad\u03bd\u03b1\u03bd \u03c0\u03bb\u03ae\u03c1\u03b7 \u03ba\u03cd\u03ba\u03bb\u03bf \u03c4\u03bf\u03c5 \u03ba\u03cd\u03bc\u03b1\u03c4\u03cc\u03c2 \u03bc\u03b1\u03c2 \u03ad\u03c0\u03c1\u03b5\u03c0\u03b5 \u03bd\u03b1 \u03c3\u03c4\u03c1\u03af\u03c8\u03bf\u03c5\u03bc\u03b5 \u03c3\u03b5 \u03b3\u03c9\u03bd\u03af\u03b1\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/f7cab61d3dd0998c89aeae012dc0cb54\/images\/img-0001.png?w=700&#038;ssl=1\" alt=\"$2\\pi $\" data-recalc-dims=\"1\" \/>.<\/p>\n<p>&nbsp;<\/p>\n<p>\u0395\u03af\u03bd\u03b1\u03b9 \u03b5\u03c0\u03af\u03c3\u03b7\u03c2 \u03b4\u03c5\u03bd\u03b1\u03c4\u03cc \u03bd\u03b1 \u03b4\u03b7\u03bc\u03b9\u03bf\u03c5\u03c1\u03b3\u03b7\u03b8\u03bf\u03cd\u03bd \u03ba\u03cd\u03bc\u03b1\u03c4\u03b1 \u03bc\u03b5 \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03b5\u03c4\u03b9\u03ba\u03ac \u03bc\u03ae\u03ba\u03b7 \u03ba\u03cd\u03bc\u03b1\u03c4\u03bf\u03c2. \u0393\u03b9\u03b1 \u03bd\u03b1 \u03b4\u03b7\u03bc\u03b9\u03bf\u03c5\u03c1\u03b3\u03ae\u03c3\u03b5\u03c4\u03b5 \u03ad\u03bd\u03b1 \u03ba\u03cd\u03bc\u03b1 \u03bc\u03b5 \u03bc\u03ae\u03ba\u03bf\u03c2 \u03ba\u03cd\u03bc\u03b1\u03c4\u03bf\u03c2\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/4429474d2e448d9c481b5426401e5340\/images\/img-0001.png?w=700&#038;ssl=1\" alt=\"$\\\u03bb\u03ac\u03bc\u03b4\u03b1 $\" data-recalc-dims=\"1\" \/>\u03c0\u03bf\u03bb\u03bb\u03b1\u03c0\u03bb\u03b1\u03c3\u03b9\u03ac\u03b6\u03b5\u03c4\u03b5 \u03c4\u03b7 \u03bc\u03b5\u03c4\u03b1\u03b2\u03bb\u03b7\u03c4\u03ae\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/4429474d2e448d9c481b5426401e5340\/images\/img-0002.png?w=700&#038;ssl=1\" alt=\"$x$\" data-recalc-dims=\"1\" \/>\u03bc\u03b5\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/4429474d2e448d9c481b5426401e5340\/images\/img-0003.png?w=700&#038;ssl=1\" alt=\"$2\\in \/\\\u03bb\u03ac\u03bc\u03b4\u03b1 $\" data-recalc-dims=\"1\" \/>\u03b3\u03b9\u03b1 \u03bd\u03b1 \u03bb\u03ac\u03b2\u03b5\u03c4\u03b5 \u03c4\u03b9\u03c2 \u03c3\u03c5\u03bd\u03b1\u03c1\u03c4\u03ae\u03c3\u03b5\u03b9\u03c2<\/p>\n<table id=\"a0000000002\" class=\"equation\" width=\"100%\" cellspacing=\"0\" cellpadding=\"7\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/4429474d2e448d9c481b5426401e5340\/images\/img-0004.png?w=700&#038;ssl=1\" alt=\"\\[ f(x)=\\sin {\\left(\\frac{2\\pi }{\\lambda }x\\right)}\\; \\; \\; \\; \\mbox{and}\\; \\; \\; \\; g(x)=\\cos {\\left(\\frac{2\\pi }{\\lambda }x\\right)}. \\]\" data-recalc-dims=\"1\" \/><\/td>\n<td><\/td>\n<td class=\"eqnnum\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u039a\u03ac\u03bd\u03bf\u03bd\u03c4\u03b1\u03c2 \u03c4\u03bf \u03bc\u03ae\u03ba\u03bf\u03c2 \u03ba\u03cd\u03bc\u03b1\u03c4\u03bf\u03c2 \u03bc\u03b9\u03ba\u03c1\u03cc\u03c4\u03b5\u03c1\u03bf, \u03bf\u03c5\u03c3\u03b9\u03b1\u03c3\u03c4\u03b9\u03ba\u03ac \u03c0\u03b9\u03ad\u03b6\u03b5\u03c4\u03b5 \u03c4\u03bf \u03ba\u03cd\u03bc\u03b1 \u03ba\u03b1\u03b9 \u03bc\u03b1\u03ba\u03c1\u03cd\u03c4\u03b5\u03c1\u03bf \u03c4\u03bf \u03c4\u03b5\u03bd\u03c4\u03ce\u03bd\u03b5\u03c4\u03b5.<\/p>\n<p>&nbsp;<\/p>\n<p>\u03a3\u03c4\u03b7\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03bc\u03b9\u03ba\u03c1\u03bf\u03b5\u03c6\u03b1\u03c1\u03bc\u03bf\u03b3\u03ae Geogebra, \u03c7\u03c1\u03b7\u03c3\u03b9\u03bc\u03bf\u03c0\u03bf\u03b9\u03ae\u03c3\u03c4\u03b5 \u03c4\u03bf \u03c1\u03c5\u03b8\u03bc\u03b9\u03c3\u03c4\u03b9\u03ba\u03cc \u03b3\u03b9\u03b1 \u03bd\u03b1 \u03b1\u03bb\u03bb\u03ac\u03be\u03b5\u03c4\u03b5 \u03c4\u03bf \u03bc\u03ae\u03ba\u03bf\u03c2 \u03ba\u03cd\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c4\u03c9\u03bd \u03b7\u03bc\u03b9\u03c4\u03bf\u03bd\u03bf\u03b5\u03b9\u03b4\u03ce\u03bd (\u03ba\u03cc\u03ba\u03ba\u03b9\u03bd\u03bf) \u03ba\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b7\u03bc\u03b9\u03c4\u03cc\u03bd\u03bf\u03c5 (\u03bc\u03c0\u03bb\u03b5) \u03ba\u03c5\u03bc\u03ac\u03c4\u03c9\u03bd.<\/p>\n<p><iframe loading=\"lazy\" title=\"\u039c\u03ae\u03ba\u03bf\u03c2 \u03ba\u03cd\u03bc\u03b1\u03c4\u03bf\u03c2 2R\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/kbeumrpa\/width\/1346\/height\/600\/border\/888888\/sfsb\/true\/smb\/false\/stb\/false\/stbh\/false\/ai\/false\/asb\/false\/sri\/false\/rc\/false\/ld\/false\/sdz\/false\/ctl\/false\" width=\"600px\" height=\"267px\" scrolling=\"no\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<h3>\u03a0\u03b7\u03b3\u03b1\u03af\u03bd\u03bf\u03bd\u03c4\u03b1\u03c2 \u03c8\u03b7\u03bb\u03cc\u03c4\u03b5\u03c1\u03b1 \u03ba\u03b1\u03b9 \u03ba\u03b1\u03c4\u03b5\u03b2\u03b1\u03af\u03bd\u03bf\u03bd\u03c4\u03b1\u03c2 \u03c7\u03b1\u03bc\u03b7\u03bb\u03cc\u03c4\u03b5\u03c1\u03b1<\/h3>\n<p>&nbsp;<\/p>\n<p>\u039f\u03b9 \u03ba\u03bf\u03c1\u03c5\u03c6\u03ad\u03c2 \u03c4\u03c9\u03bd \u03ba\u03c5\u03bc\u03ac\u03c4\u03c9\u03bd \u03c0\u03bf\u03c5 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b7\u03bc\u03b9\u03bf\u03c5\u03c1\u03b3\u03ae\u03c3\u03b5\u03b9 \u03bc\u03ad\u03c7\u03c1\u03b9 \u03c4\u03ce\u03c1\u03b1 \u03ad\u03c7\u03bf\u03c5\u03bd \u03c4\u03b7\u03bd \u03c4\u03b9\u03bc\u03ae 1 \u03ba\u03b1\u03b9 \u03bf\u03b9 \u03b3\u03bf\u03cd\u03c1\u03bd\u03b5\u03c2 \u03c4\u03b7\u03bd \u03c4\u03b9\u03bc\u03ae -1. \u0393\u03b9\u03b1 \u03bd\u03b1 \u03b4\u03b7\u03bc\u03b9\u03bf\u03c5\u03c1\u03b3\u03ae\u03c3\u03b5\u03c4\u03b5 \u03ad\u03bd\u03b1 \u03ba\u03cd\u03bc\u03b1 \u03c0\u03bf\u03c5 \u03ad\u03c7\u03b5\u03b9 \u03c5\u03c8\u03b7\u03bb\u03cc\u03c4\u03b5\u03c1\u03b5\u03c2 \u03ba\u03bf\u03c1\u03c5\u03c6\u03ad\u03c2 \u03ba\u03b1\u03b9 \u03c7\u03b1\u03bc\u03b7\u03bb\u03cc\u03c4\u03b5\u03c1\u03b5\u03c2 \u03ba\u03bf\u03b9\u03bb\u03cc\u03c4\u03b7\u03c4\u03b5\u03c2, \u03b1\u03c0\u03bb\u03ce\u03c2 \u03c0\u03bf\u03bb\u03bb\u03b1\u03c0\u03bb\u03b1\u03c3\u03b9\u03ac\u03c3\u03c4\u03b5 \u03bf\u03bb\u03cc\u03ba\u03bb\u03b7\u03c1\u03b7 \u03c4\u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u03bc\u03b5 \u03bc\u03b9\u03b1 \u03c3\u03c4\u03b1\u03b8\u03b5\u03c1\u03ac\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/d19b1b9dddd914e19eb6f42ab965aaa5\/images\/img-0001.png?w=700&#038;ssl=1\" alt=\"$A$\" data-recalc-dims=\"1\" \/>\u03b3\u03b9\u03b1 \u03bd\u03b1 \u03c0\u03ac\u03c1\u03b5\u03c4\u03b5<\/p>\n<table id=\"a0000000002\" class=\"equation\" width=\"100%\" cellspacing=\"0\" cellpadding=\"7\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/d19b1b9dddd914e19eb6f42ab965aaa5\/images\/img-0002.png?w=700&#038;ssl=1\" alt=\"\\[ f(x)=A\\sin {\\left(\\frac{2\\pi }{\\lambda }x\\right)}\\; \\; \\; \\; \\mbox{and}\\; \\; \\; \\; g(x)=A\\cos {\\left(\\frac{2\\pi }{\\lambda }x\\right)}. \\]\" data-recalc-dims=\"1\" \/><\/td>\n<td><\/td>\n<td class=\"eqnnum\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u0391\u03c5\u03c4\u03ae \u03b7 \u03c3\u03c4\u03b1\u03b8\u03b5\u03c1\u03ac \u03bf\u03bd\u03bf\u03bc\u03ac\u03b6\u03b5\u03c4\u03b1\u03b9\u00a0<em>\u03c0\u03bb\u03ac\u03c4\u03bf\u03c2<\/em>\u00a0\u03c4\u03bf\u03c5 \u03ba\u03cd\u03bc\u03b1\u03c4\u03bf\u03c2.<\/p>\n<p>&nbsp;<\/p>\n<p>\u03a3\u03c4\u03b7\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03bc\u03b9\u03ba\u03c1\u03bf\u03b5\u03c6\u03b1\u03c1\u03bc\u03bf\u03b3\u03ae Geogebra \u03c7\u03c1\u03b7\u03c3\u03b9\u03bc\u03bf\u03c0\u03bf\u03b9\u03ae\u03c3\u03c4\u03b5 \u03c4\u03b1 \u03c1\u03c5\u03b8\u03bc\u03b9\u03c3\u03c4\u03b9\u03ba\u03ac \u03b3\u03b9\u03b1 \u03bd\u03b1 \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03bf\u03c0\u03bf\u03b9\u03ae\u03c3\u03b5\u03c4\u03b5 \u03c4\u03bf \u03c0\u03bb\u03ac\u03c4\u03bf\u03c2\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/2f751c9fcca1a311489bc47800dfb670\/images\/img-0001.png?w=700&#038;ssl=1\" alt=\"$A$\" data-recalc-dims=\"1\" \/>\u03ba\u03b1\u03b9 \u03c4\u03bf \u03bc\u03ae\u03ba\u03bf\u03c2 \u03ba\u03cd\u03bc\u03b1\u03c4\u03bf\u03c2\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/2f751c9fcca1a311489bc47800dfb670\/images\/img-0002.png?w=700&#038;ssl=1\" alt=\"$\\\u03bb\u03ac\u03bc\u03b4\u03b1 $\" data-recalc-dims=\"1\" \/>\u03b3\u03b9\u03b1 \u03c4\u03bf \u03b7\u03bc\u03b9\u03c4\u03bf\u03bd\u03bf\u03b5\u03b9\u03b4\u03ad\u03c2 \u03ba\u03cd\u03bc\u03b1 (\u03ba\u03cc\u03ba\u03ba\u03b9\u03bd\u03bf) \u03ba\u03b1\u03b9 \u03c4\u03bf \u03c3\u03c5\u03bd\u03b7\u03bc\u03b9\u03c4\u03bf\u03bd\u03b9\u03ba\u03cc \u03ba\u03cd\u03bc\u03b1 (\u03bc\u03c0\u03bb\u03b5).<\/p>\n<p><iframe loading=\"lazy\" title=\"\u03bc\u03ae\u03ba\u03bf\u03c2 \u03ba\u03cd\u03bc\u03b1\u03c4\u03bf\u03c2 3R\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/enjpfyu5\/width\/1346\/height\/600\/border\/888888\/sfsb\/true\/smb\/false\/stb\/false\/stbh\/false\/ai\/false\/asb\/false\/sri\/false\/rc\/false\/ld\/false\/sdz\/false\/ctl\/false\" width=\"600px\" height=\"267px\" scrolling=\"no\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<h3>\u0395\u03c0\u03b9\u03c4\u03ac\u03c7\u03c5\u03bd\u03c3\u03b7 \u03ba\u03b1\u03b9 \u03b5\u03c0\u03b9\u03b2\u03c1\u03ac\u03b4\u03c5\u03bd\u03c3\u03b7<\/h3>\n<p>&nbsp;<\/p>\n<p>\u039c\u03ad\u03c7\u03c1\u03b9 \u03c3\u03c4\u03b9\u03b3\u03bc\u03ae\u03c2 \u03c4\u03b1 \u03ba\u03cd\u03bc\u03b1\u03c4\u03b1 \u03c0\u03bf\u03c5 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b7\u03bc\u03b9\u03bf\u03c5\u03c1\u03b3\u03ae\u03c3\u03b5\u03b9 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b1\u03ba\u03af\u03bd\u03b7\u03c4\u03b1: \u03b4\u03b5\u03bd \u03b1\u03bb\u03bb\u03ac\u03b6\u03bf\u03c5\u03bd \u03bc\u03b5 \u03c4\u03b7\u03bd \u03c0\u03ac\u03c1\u03bf\u03b4\u03bf \u03c4\u03bf\u03c5 \u03c7\u03c1\u03cc\u03bd\u03bf\u03c5. \u03a9\u03c3\u03c4\u03cc\u03c3\u03bf, \u03b5\u03af\u03bd\u03b1\u03b9 \u03b5\u03c0\u03af\u03c3\u03b7\u03c2 \u03b4\u03c5\u03bd\u03b1\u03c4\u03cc \u03bd\u03b1 \u03b4\u03b7\u03bc\u03b9\u03bf\u03c5\u03c1\u03b3\u03b7\u03b8\u03bf\u03cd\u03bd \u03ba\u03cd\u03bc\u03b1\u03c4\u03b1 \u03c0\u03bf\u03c5 \u03c4\u03b1\u03be\u03b9\u03b4\u03b5\u03cd\u03bf\u03c5\u03bd \u03ba\u03b1\u03c4\u03ac \u03bc\u03ae\u03ba\u03bf\u03c2. \u0397 \u03ba\u03c5\u03bc\u03b1\u03c4\u03b9\u03ba\u03ae \u03bc\u03b1\u03c2 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u03c4\u03ce\u03c1\u03b1 \u03b8\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u03b4\u03cd\u03bf \u03bc\u03b5\u03c4\u03b1\u03b2\u03bb\u03b7\u03c4\u03ce\u03bd\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/953d2339d460430195fbb0cf1b88d898\/images\/img-0001.png?w=700&#038;ssl=1\" alt=\"$x$\" data-recalc-dims=\"1\" \/>\u03ba\u03b1\u03b9\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/953d2339d460430195fbb0cf1b88d898\/images\/img-0002.png?w=700&#038;ssl=1\" alt=\"$t.$\" data-recalc-dims=\"1\" \/>\u03cc\u03c0\u03c9\u03c2 \u03c0\u03c1\u03b9\u03bd\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/953d2339d460430195fbb0cf1b88d898\/images\/img-0001.png?w=700&#038;ssl=1\" alt=\"$x$\" data-recalc-dims=\"1\" \/>\u03b1\u03bd\u03c4\u03b9\u03c0\u03c1\u03bf\u03c3\u03c9\u03c0\u03b5\u03cd\u03b5\u03b9 \u03c4\u03b7 \u03b8\u03ad\u03c3\u03b7 \u03c3\u03c4\u03bf\u03bd \u03bf\u03c1\u03b9\u03b6\u03cc\u03bd\u03c4\u03b9\u03bf \u03ac\u03be\u03bf\u03bd\u03b1, \u03b5\u03bd\u03ce\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/953d2339d460430195fbb0cf1b88d898\/images\/img-0003.png?w=700&#038;ssl=1\" alt=\"$t$\" data-recalc-dims=\"1\" \/>\u03c5\u03c0\u03bf\u03b4\u03b7\u03bb\u03ce\u03bd\u03b5\u03b9 \u03c7\u03c1\u03cc\u03bd\u03bf. \u0391\u03c2 \u03c5\u03c0\u03bf\u03b8\u03ad\u03c3\u03bf\u03c5\u03bc\u03b5 \u03cc\u03c4\u03b9 \u03b8\u03ad\u03bb\u03b5\u03c4\u03b5 \u03c4\u03bf \u03ba\u03cd\u03bc\u03b1 \u03c3\u03b1\u03c2 \u03bd\u03b1 \u03ba\u03b9\u03bd\u03b5\u03af\u03c4\u03b1\u03b9 \u03bc\u03b5 \u03c4\u03b1\u03c7\u03cd\u03c4\u03b7\u03c4\u03b1 \u03c4\u03c9\u03bd\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/953d2339d460430195fbb0cf1b88d898\/images\/img-0004.png?w=700&#038;ssl=1\" alt=\"$v.$\" data-recalc-dims=\"1\" \/>\u03c3\u03c5\u03bd\u03b1\u03c1\u03c4\u03ae\u03c3\u03b5\u03c9\u03bd<\/p>\n<table id=\"a0000000002\" class=\"equation\" width=\"100%\" cellspacing=\"0\" cellpadding=\"7\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/953d2339d460430195fbb0cf1b88d898\/images\/img-0005.png?w=700&#038;ssl=1\" alt=\"\\[ f(x,t)=A\\sin {\\left(\\frac{2\\pi }{\\lambda }x-vt\\right)}\\; \\; \\; \\; \\mbox{and}\\; \\; \\; \\; g(x,t)=A\\cos {\\left(\\frac{2\\pi }{\\lambda }x-vt\\right)}. \\]\" data-recalc-dims=\"1\" \/><\/td>\n<td><\/td>\n<td class=\"eqnnum\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u03c0\u03b1\u03c1\u03ac\u03b3\u03bf\u03c5\u03bd \u03c4\u03ad\u03c4\u03bf\u03b9\u03b1 \u03ba\u03b9\u03bd\u03bf\u03cd\u03bc\u03b5\u03bd\u03b1 \u03ba\u03cd\u03bc\u03b1\u03c4\u03b1. \u0395\u03ac\u03bd \u03b4\u03b9\u03b1\u03c4\u03b7\u03c1\u03b5\u03af\u03c4\u03b5 \u03c4\u03b7 \u03bc\u03b5\u03c4\u03b1\u03b2\u03bb\u03b7\u03c4\u03ae \u03c7\u03c1\u03cc\u03bd\u03bf\u03c5\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/953d2339d460430195fbb0cf1b88d898\/images\/img-0003.png?w=700&#038;ssl=1\" alt=\"$t$\" data-recalc-dims=\"1\" \/>\u03c3\u03c4\u03b1\u03b8\u03b5\u03c1\u03ae, \u03c4\u03cc\u03c4\u03b5 \u03bf\u03c5\u03c3\u03b9\u03b1\u03c3\u03c4\u03b9\u03ba\u03ac \u03b2\u03bb\u03ad\u03c0\u03b5\u03c4\u03b5 \u03ad\u03bd\u03b1 \u03c3\u03c4\u03b9\u03b3\u03bc\u03b9\u03cc\u03c4\u03c5\u03c0\u03bf \u03c3\u03c4\u03bf \u03c7\u03c1\u03cc\u03bd\u03bf, \u03b4\u03af\u03bd\u03bf\u03bd\u03c4\u03ac\u03c2 \u03c3\u03b1\u03c2 \u03ad\u03bd\u03b1 \u03b1\u03ba\u03af\u03bd\u03b7\u03c4\u03bf \u03ba\u03cd\u03bc\u03b1 \u03cc\u03c0\u03c9\u03c2 \u03b1\u03c5\u03c4\u03ac \u03c0\u03bf\u03c5 \u03b5\u03af\u03c7\u03b1\u03bc\u03b5 \u03c0\u03b1\u03c1\u03b1\u03c0\u03ac\u03bd\u03c9. \u0395\u03ac\u03bd \u03b4\u03b9\u03b1\u03c4\u03b7\u03c1\u03b5\u03af\u03c4\u03b5 \u03c3\u03c4\u03b1\u03b8\u03b5\u03c1\u03ae \u03c4\u03b7\u03bd \u03c4\u03bf\u03c0\u03bf\u03b8\u03b5\u03c3\u03af\u03b1 \u03c3\u03b1\u03c2\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/953d2339d460430195fbb0cf1b88d898\/images\/img-0001.png?w=700&#038;ssl=1\" alt=\"$x$\" data-recalc-dims=\"1\" \/>\u03c3\u03c4\u03bf\u03bd\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/953d2339d460430195fbb0cf1b88d898\/images\/img-0001.png?w=700&#038;ssl=1\" alt=\"$x$\" data-recalc-dims=\"1\" \/>\u03ac\u03be\u03bf\u03bd\u03b1 -, \u03c4\u03cc\u03c4\u03b5 \u03b2\u03bb\u03ad\u03c0\u03b5\u03c4\u03b5 \u03c4\u03b7\u03bd \u03b1\u03bd\u03c4\u03af\u03c3\u03c4\u03bf\u03b9\u03c7\u03b7\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/953d2339d460430195fbb0cf1b88d898\/images\/img-0006.png?w=700&#038;ssl=1\" alt=\"$y$\" data-recalc-dims=\"1\" \/>\u03bc\u03b5\u03c4\u03b1\u03b2\u03bb\u03b7\u03c4\u03ae &#8211; \u03ba\u03b1\u03b8\u03ce\u03c2 \u03ba\u03b9\u03bd\u03b5\u03af\u03c4\u03b1\u03b9 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03c0\u03ac\u03bd\u03c9 \u03ba\u03b1\u03b9 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ba\u03ac\u03c4\u03c9 \u03bc\u03b5 \u03c4\u03b7\u03bd \u03c0\u03ac\u03c1\u03bf\u03b4\u03bf \u03c4\u03bf\u03c5 \u03c7\u03c1\u03cc\u03bd\u03bf\u03c5. \u0395\u03af\u03bd\u03b1\u03b9 \u03bb\u03af\u03b3\u03bf \u03c3\u03b1\u03bd \u03bd\u03b1 \u03c0\u03b1\u03c1\u03b1\u03ba\u03bf\u03bb\u03bf\u03c5\u03b8\u03b5\u03af\u03c2 \u03ad\u03bd\u03b1 \u03c3\u03c5\u03b3\u03ba\u03b5\u03ba\u03c1\u03b9\u03bc\u03ad\u03bd\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03c3\u03c4\u03b7\u03bd \u03b5\u03c0\u03b9\u03c6\u03ac\u03bd\u03b5\u03b9\u03b1 \u03bc\u03b9\u03b1\u03c2 \u03bb\u03af\u03bc\u03bd\u03b7\u03c2 \u03bd\u03b1 \u03ba\u03b9\u03bd\u03b5\u03af\u03c4\u03b1\u03b9 \u03c0\u03ac\u03bd\u03c9-\u03ba\u03ac\u03c4\u03c9 \u03ba\u03b1\u03b8\u03ce\u03c2 \u03ba\u03c5\u03bc\u03b1\u03c4\u03af\u03b6\u03bf\u03c5\u03bd \u03c4\u03b1 \u03ba\u03cd\u03bc\u03b1\u03c4\u03b1.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>\u03a3\u03c4\u03b7\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03bc\u03b9\u03ba\u03c1\u03bf\u03b5\u03c6\u03b1\u03c1\u03bc\u03bf\u03b3\u03ae Geogebra \u03c7\u03c1\u03b7\u03c3\u03b9\u03bc\u03bf\u03c0\u03bf\u03b9\u03ae\u03c3\u03c4\u03b5 \u03c4\u03bf \u03c1\u03c5\u03b8\u03bc\u03b9\u03c3\u03c4\u03b9\u03ba\u03cc \u03b3\u03b9\u03b1 \u03bd\u03b1 \u03b1\u03bb\u03bb\u03ac\u03be\u03b5\u03c4\u03b5 \u03c4\u03b7\u03bd \u03c4\u03b1\u03c7\u03cd\u03c4\u03b7\u03c4\u03b1. \u0397 \u03c1\u03cd\u03b8\u03bc\u03b9\u03c3\u03b7 \u03c4\u03b7\u03c2 \u03c4\u03b1\u03c7\u03cd\u03c4\u03b7\u03c4\u03b1\u03c2 \u03c3\u03c4\u03bf 0 (\u03ae \u03c4\u03bf \u03c0\u03ac\u03c4\u03b7\u03bc\u03b1 \u03c4\u03bf\u03c5 \u03ba\u03bf\u03c5\u03bc\u03c0\u03b9\u03bf\u03cd \u03c0\u03b1\u03cd\u03c3\u03b7\u03c2) \u03b1\u03bd\u03c4\u03b9\u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u03af \u03c3\u03c4\u03bf\u03bd \u03c7\u03c1\u03cc\u03bd\u03bf \u03b4\u03b9\u03b1\u03ba\u03bf\u03c0\u03ae\u03c2, \u03b5\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u03b4\u03b9\u03b1\u03c4\u03b7\u03c1\u03b5\u03af\u03c4\u03b5\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/a6301a7ba8cf64773345779e825e048f\/images\/img-0001.png?w=700&#038;ssl=1\" alt=\"$t$\" data-recalc-dims=\"1\" \/>\u03c3\u03c4\u03b1\u03b8\u03b5\u03c1\u03ae \u03c4\u03b7 \u03bc\u03b5\u03c4\u03b1\u03b2\u03bb\u03b7\u03c4\u03ae \u03c4\u03bf\u03c5 \u03c7\u03c1\u03cc\u03bd\u03bf\u03c5. \u03a3\u03b5 \u03b1\u03c5\u03c4\u03ae\u03bd \u03c4\u03b7\u03bd \u03c0\u03b5\u03c1\u03af\u03c0\u03c4\u03c9\u03c3\u03b7 \u03b2\u03bb\u03ad\u03c0\u03b5\u03c4\u03b5 \u03ad\u03bd\u03b1 \u03c3\u03c4\u03b9\u03b3\u03bc\u03b9\u03cc\u03c4\u03c5\u03c0\u03bf \u03c3\u03c4\u03bf \u03c7\u03c1\u03cc\u03bd\u03bf \u03c0\u03bf\u03c5 \u03bc\u03bf\u03b9\u03ac\u03b6\u03b5\u03b9 \u03bc\u03b5 \u03ad\u03bd\u03b1 \u03c3\u03c5\u03bd\u03b7\u03b8\u03b9\u03c3\u03bc\u03ad\u03bd\u03bf \u03b7\u03bc\u03b9\u03c4\u03bf\u03bd\u03bf\u03b5\u03b9\u03b4\u03ad\u03c2 (\u03ae \u03c3\u03c5\u03bd\u03b7\u03bc\u03b9\u03c4\u03bf\u03bd\u03b9\u03ba\u03cc) \u03ba\u03cd\u03bc\u03b1, \u03bc\u03cc\u03bd\u03bf \u03af\u03c3\u03c9\u03c2 \u03bc\u03b5\u03c4\u03b1\u03c4\u03bf\u03c0\u03b9\u03c3\u03bc\u03ad\u03bd\u03bf \u03ba\u03b1\u03c4\u03ac \u03bc\u03ae\u03ba\u03bf\u03c2 \u03c4\u03bf\u03c5 \u03bf\u03c1\u03b9\u03b6\u03cc\u03bd\u03c4\u03b9\u03bf\u03c5 \u03ac\u03be\u03bf\u03bd\u03b1 \u03ba\u03b1\u03c4\u03ac \u03ba\u03ac\u03c0\u03bf\u03b9\u03b1 \u03b1\u03c0\u03cc\u03c3\u03c4\u03b1\u03c3\u03b7. \u0397 \u03b5\u03c3\u03c4\u03af\u03b1\u03c3\u03b7 \u03c3\u03c4\u03b7\u03bd \u03bc\u03c0\u03bb\u03b5 \u03ba\u03bf\u03c5\u03ba\u03ba\u03af\u03b4\u03b1 \u03b1\u03bd\u03c4\u03b9\u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u03af \u03c3\u03c4\u03b7 \u03b4\u03b9\u03b1\u03c4\u03ae\u03c1\u03b7\u03c3\u03b7 \u03c4\u03b7\u03c2 \u03bc\u03b5\u03c4\u03b1\u03b2\u03bb\u03b7\u03c4\u03ae\u03c2\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/a6301a7ba8cf64773345779e825e048f\/images\/img-0002.png?w=700&#038;ssl=1\" alt=\"$x$\" data-recalc-dims=\"1\" \/>\u03c3\u03c4\u03b1\u03b8\u03b5\u03c1\u03ae. (\u0394\u03b5\u03bd \u03c3\u03b1\u03c2 \u03b4\u03af\u03bd\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b7\u03bd \u03b5\u03c0\u03b9\u03bb\u03bf\u03b3\u03ae \u03bd\u03b1 \u03b1\u03bb\u03bb\u03ac\u03be\u03b5\u03c4\u03b5 \u03c4\u03bf \u03bc\u03ae\u03ba\u03bf\u03c2 \u03ba\u03cd\u03bc\u03b1\u03c4\u03bf\u03c2 \u03ba\u03b1\u03b9 \u03c4\u03bf \u03c0\u03bb\u03ac\u03c4\u03bf\u03c2 \u03b5\u03b4\u03ce, \u03ba\u03b1\u03b8\u03ce\u03c2 \u03b7 \u03c5\u03c0\u03b5\u03c1\u03b2\u03bf\u03bb\u03b9\u03ba\u03ae \u03b5\u03c0\u03b9\u03bb\u03bf\u03b3\u03ae \u03bc\u03c0\u03bf\u03c1\u03b5\u03af \u03bd\u03b1 \u03c0\u03c1\u03bf\u03ba\u03b1\u03bb\u03ad\u03c3\u03b5\u03b9 \u03c3\u03cd\u03b3\u03c7\u03c5\u03c3\u03b7!)<\/p>\n<p>&nbsp;<\/p>\n<p><iframe loading=\"lazy\" title=\"WavetimeMF\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/rfyatswn\/width\/782\/height\/478\/border\/888888\/sfsb\/true\/smb\/false\/stb\/false\/stbh\/false\/ai\/false\/asb\/false\/sri\/false\/rc\/false\/ld\/false\/sdz\/false\/ctl\/false\" width=\"600px\" height=\"478px\" scrolling=\"no\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<p>\u039f \u03bb\u03cc\u03b3\u03bf\u03c2 \u03c0\u03bf\u03c5 \u03c4\u03bf \u03b5\u03c0\u03b9\u03c7\u03b5\u03af\u03c1\u03b7\u03bc\u03b1 \u03c4\u03c9\u03bd \u03c3\u03c5\u03bd\u03b1\u03c1\u03c4\u03ae\u03c3\u03b5\u03ce\u03bd \u03bc\u03b1\u03c2 \u03c4\u03ce\u03c1\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/58377f8f0a9b1fcb7c0f678279d0ee51\/images\/img-0001.png?w=700&#038;ssl=1\" alt=\"$x-vt$\" data-recalc-dims=\"1\" \/>\u03cc\u03c4\u03b9 \u03c3\u03b5 \u03bc\u03b9\u03b1 \u03c7\u03c1\u03bf\u03bd\u03b9\u03ba\u03ae \u03c0\u03b5\u03c1\u03af\u03bf\u03b4\u03bf \u03bc\u03ae\u03ba\u03bf\u03c5\u03c2\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/58377f8f0a9b1fcb7c0f678279d0ee51\/images\/img-0002.png?w=700&#038;ssl=1\" alt=\"$t$\" data-recalc-dims=\"1\" \/>\u03ad\u03bd\u03b1 \u03ba\u03cd\u03bc\u03b1 \u03c0\u03bf\u03c5 \u03c4\u03b1\u03be\u03b9\u03b4\u03b5\u03cd\u03b5\u03b9 \u03bc\u03b5 \u03c4\u03b1\u03c7\u03cd\u03c4\u03b7\u03c4\u03b1\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/58377f8f0a9b1fcb7c0f678279d0ee51\/images\/img-0003.png?w=700&#038;ssl=1\" alt=\"$\u03c3\u03b5 $\" data-recalc-dims=\"1\" \/>\u03b8\u03b1 \u03ad\u03c7\u03b5\u03b9 \u03b4\u03b9\u03b1\u03bd\u03cd\u03c3\u03b5\u03b9 \u03b1\u03c0\u03cc\u03c3\u03c4\u03b1\u03c3\u03b7\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/58377f8f0a9b1fcb7c0f678279d0ee51\/images\/img-0004.png?w=700&#038;ssl=1\" alt=\"$vt.$\" data-recalc-dims=\"1\" \/>\u03a4\u03bf \u03cd\u03c8\u03bf\u03c2 \u03c4\u03bf\u03c5 \u03ba\u03cd\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c3\u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/58377f8f0a9b1fcb7c0f678279d0ee51\/images\/img-0005.png?w=700&#038;ssl=1\" alt=\"$x$\" data-recalc-dims=\"1\" \/>\u03ba\u03b1\u03c4\u03ac \u03bc\u03ae\u03ba\u03bf\u03c2 \u03c4\u03bf\u03c5\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/58377f8f0a9b1fcb7c0f678279d0ee51\/images\/img-0005.png?w=700&#038;ssl=1\" alt=\"$x$\" data-recalc-dims=\"1\" \/>\u03ac\u03be\u03bf\u03bd\u03b1 &#8211; \u03c4\u03b7 \u03c3\u03c4\u03b9\u03b3\u03bc\u03ae\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/58377f8f0a9b1fcb7c0f678279d0ee51\/images\/img-0002.png?w=700&#038;ssl=1\" alt=\"$t$\" data-recalc-dims=\"1\" \/>\u03b8\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b5\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u03c4\u03bf \u03af\u03b4\u03b9\u03bf \u03bc\u03b5 \u03c4\u03bf \u03cd\u03c8\u03bf\u03c2 \u03c4\u03b7 \u03c7\u03c1\u03bf\u03bd\u03b9\u03ba\u03ae \u03c3\u03c4\u03b9\u03b3\u03bc\u03ae 0 \u03c3\u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/58377f8f0a9b1fcb7c0f678279d0ee51\/images\/img-0006.png?w=700&#038;ssl=1\" alt=\"$x-vt,$\" data-recalc-dims=\"1\" \/>\u03b3\u03b9\u03b1\u03c4\u03af \u03c4\u03cc\u03c3\u03bf \u03bc\u03b1\u03ba\u03c1\u03b9\u03ac \u03ad\u03c7\u03b5\u03b9 \u03b4\u03b9\u03b1\u03bd\u03cd\u03c3\u03b5\u03b9 \u03c4\u03bf \u03ba\u03cd\u03bc\u03b1 \u03c3\u03c4\u03bf \u03c7\u03c1\u03cc\u03bd\u03bf\u00a0<img decoding=\"async\" class=\"math gen\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/MI\/58377f8f0a9b1fcb7c0f678279d0ee51\/images\/img-0002.png?w=700&#038;ssl=1\" alt=\"$t$\" data-recalc-dims=\"1\" \/>.<\/p>\n<p>&nbsp;<\/p>\n<h3>\u0391\u03c0\u03bf\u03ba\u03c9\u03b4\u03b9\u03ba\u03bf\u03c0\u03bf\u03af\u03b7\u03c3\u03b7 \u03bc\u03b7\u03bd\u03c5\u03bc\u03ac\u03c4\u03c9\u03bd<\/h3>\n<p>\u0395\u03ba \u03c0\u03c1\u03ce\u03c4\u03b7\u03c2 \u03cc\u03c8\u03b5\u03c9\u03c2, \u03b1\u03c5\u03c4\u03ac \u03c4\u03b1 \u03b7\u03bc\u03b9\u03c4\u03bf\u03bd\u03bf\u03b5\u03b9\u03b4\u03ae \u03ba\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b7\u03bc\u03b9\u03c4\u03bf\u03bd\u03bf\u03b5\u03b9\u03b4\u03ae \u03ba\u03cd\u03bc\u03b1\u03c4\u03b1 \u03c0\u03bf\u03c5 \u03b4\u03b7\u03bc\u03b9\u03bf\u03c5\u03c1\u03b3\u03ae\u03c3\u03b1\u03bc\u03b5 \u03c6\u03b1\u03af\u03bd\u03bf\u03bd\u03c4\u03b1\u03b9 \u03c0\u03bf\u03bb\u03cd \u03c4\u03ad\u03bb\u03b5\u03b9\u03b1 \u03b3\u03b9\u03b1 \u03bd\u03b1 \u03c3\u03b1\u03c2 \u03c0\u03bf\u03c5\u03bd \u03c0\u03bf\u03bb\u03bb\u03ac \u03b3\u03b9\u03b1 \u03c4\u03b1 \u03ba\u03cd\u03bc\u03b1\u03c4\u03b1 \u03c0\u03bf\u03c5 \u03c3\u03c5\u03bd\u03b1\u03bd\u03c4\u03ac\u03bc\u03b5 \u03c3\u03c4\u03b7\u03bd \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03ae \u03b6\u03c9\u03ae. \u0391\u03bb\u03bb\u03ac \u03bc\u03c0\u03bf\u03c1\u03b5\u03af\u03c4\u03b5 \u03bd\u03b1 \u03b1\u03ba\u03bf\u03cd\u03c3\u03b5\u03c4\u03b5 \u03ad\u03bd\u03b1 \u03c3\u03b5 \u03b4\u03c1\u03ac\u03c3\u03b7 \u03b1\u03bd \u03c7\u03c4\u03c5\u03c0\u03ae\u03c3\u03b5\u03c4\u03b5 \u03ad\u03bd\u03b1 \u03c0\u03b9\u03c1\u03bf\u03cd\u03bd\u03b9 \u03c3\u03c5\u03bd\u03c4\u03bf\u03bd\u03b9\u03c3\u03bc\u03bf\u03cd. \u039f \u03ae\u03c7\u03bf\u03c2 \u03c0\u03bf\u03c5 \u03b1\u03ba\u03bf\u03cd\u03c4\u03b5 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03bf \u03b1\u03c0\u03bf\u03c4\u03ad\u03bb\u03b5\u03c3\u03bc\u03b1 \u03c4\u03c9\u03bd \u03b4\u03bf\u03bd\u03ae\u03c3\u03b5\u03c9\u03bd \u03c4\u03bf\u03c5 \u03c4\u03c5\u03bc\u03c0\u03ac\u03bd\u03bf\u03c5 \u03c4\u03bf\u03c5 \u03b1\u03c5\u03c4\u03b9\u03bf\u03cd \u03c3\u03b1\u03c2, \u03c0\u03bf\u03c5 \u03b4\u03b9\u03b5\u03b3\u03b5\u03af\u03c1\u03bf\u03bd\u03c4\u03b1\u03b9 \u03b1\u03c0\u03cc \u03c4\u03bf \u03b7\u03c7\u03b7\u03c4\u03b9\u03ba\u03cc \u03ba\u03cd\u03bc\u03b1 \u03b1\u03c0\u03cc \u03c4\u03bf \u03c0\u03b9\u03c1\u03bf\u03cd\u03bd\u03b9 \u03c3\u03c5\u03bd\u03c4\u03bf\u03bd\u03b9\u03c3\u03bc\u03bf\u03cd \u03c0\u03bf\u03c5 \u03c4\u03b1\u03be\u03b9\u03b4\u03b5\u03cd\u03b5\u03b9 \u03c3\u03c4\u03bf\u03bd \u03b1\u03ad\u03c1\u03b1. \u0393\u03b9\u03b1 \u03bc\u03b9\u03b1 \u03b4\u03b9\u03c7\u03ac\u03bb\u03b1 \u03c3\u03c5\u03bd\u03c4\u03bf\u03bd\u03b9\u03c3\u03bc\u03bf\u03cd, \u03b1\u03bd \u03c3\u03c7\u03b5\u03b4\u03b9\u03ac\u03b6\u03b1\u03c4\u03b5 \u03c4\u03b7\u03bd \u03ad\u03bd\u03c4\u03b1\u03c3\u03b7 \u03ae \u03c4\u03b7\u03bd \u03c0\u03af\u03b5\u03c3\u03b7 \u03b1\u03c5\u03c4\u03ae\u03c2 \u03c4\u03b7\u03c2 \u03b4\u03cc\u03bd\u03b7\u03c3\u03b7\u03c2 \u03bc\u03b5 \u03c4\u03b7\u03bd \u03c0\u03ac\u03c1\u03bf\u03b4\u03bf \u03c4\u03bf\u03c5 \u03c7\u03c1\u03cc\u03bd\u03bf\u03c5, \u03b8\u03b1 \u03b2\u03bb\u03ad\u03c0\u03b1\u03c4\u03b5 \u03ad\u03bd\u03b1 \u03c4\u03ad\u03bb\u03b5\u03b9\u03bf \u03b7\u03bc\u03b9\u03c4\u03bf\u03bd\u03bf\u03b5\u03b9\u03b4\u03ad\u03c2 \u03ba\u03cd\u03bc\u03b1 \u03c3\u03b5 \u03b4\u03c1\u03ac\u03c3\u03b7.<\/p>\n<div class=\"centreimage\">\n<p><img decoding=\"async\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/content\/sites\/plus.maths.org\/files\/articles\/2017\/carola\/tuningfork.png?w=700&#038;ssl=1\" alt=\"\u0397\u03c7\u03b7\u03c4\u03b9\u03ba\u03cc \u03ba\u03cd\u03bc\u03b1 \u03b5\u03bd\u03cc\u03c2 \u03c0\u03b9\u03c1\u03bf\u03c5\u03bd\u03b9\u03bf\u03cd \u03c3\u03c5\u03bd\u03c4\u03bf\u03bd\u03b9\u03c3\u03bc\u03bf\u03cd\" data-recalc-dims=\"1\" \/><img decoding=\"async\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/content\/sites\/plus.maths.org\/files\/articles\/2017\/carola\/speech.png?w=700&#038;ssl=1\" alt=\"\u0397\u03c7\u03b7\u03c4\u03b9\u03ba\u03cc \u03ba\u03cd\u03bc\u03b1 \u03b1\u03bd\u03b8\u03c1\u03ce\u03c0\u03b9\u03bd\u03b7\u03c2 \u03bf\u03bc\u03b9\u03bb\u03af\u03b1\u03c2\" data-recalc-dims=\"1\" \/>\u03a4\u03bf \u03b7\u03c7\u03b7\u03c4\u03b9\u03ba\u03cc \u03ba\u03cd\u03bc\u03b1 \u03b1\u03c0\u03cc \u03ad\u03bd\u03b1 \u03c0\u03b9\u03c1\u03bf\u03cd\u03bd\u03b9 \u03c3\u03c5\u03bd\u03c4\u03bf\u03bd\u03b9\u03c3\u03bc\u03bf\u03cd (\u03c0\u03ac\u03bd\u03c9), \u03c3\u03b5 \u03c3\u03cd\u03b3\u03ba\u03c1\u03b9\u03c3\u03b7 \u03bc\u03b5 \u03b1\u03c5\u03c4\u03cc \u03c4\u03b7\u03c2 \u03b1\u03bd\u03b8\u03c1\u03ce\u03c0\u03b9\u03bd\u03b7\u03c2 \u03bf\u03bc\u03b9\u03bb\u03af\u03b1\u03c2 (\u03ba\u03ac\u03c4\u03c9).<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>\u038c\u03c0\u03c9\u03c2 \u03bc\u03c0\u03bf\u03c1\u03b5\u03af\u03c4\u03b5 \u03bd\u03b1 \u03b4\u03b5\u03af\u03c4\u03b5 \u03c4\u03bf \u03b7\u03c7\u03b7\u03c4\u03b9\u03ba\u03cc \u03ba\u03cd\u03bc\u03b1 \u03ba\u03ac\u03c4\u03b9 \u03c3\u03b1\u03bd \u03bf\u03bc\u03b9\u03bb\u03af\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03b9\u03bf \u03c0\u03b5\u03c1\u03af\u03c0\u03bb\u03bf\u03ba\u03bf. \u0391\u03bb\u03bb\u03ac \u03ba\u03ac\u03b8\u03b5 \u03b7\u03c7\u03b7\u03c4\u03b9\u03ba\u03cc \u03ba\u03cd\u03bc\u03b1, \u03bc\u03ac\u03bb\u03b9\u03c3\u03c4\u03b1 \u03ba\u03ac\u03b8\u03b5 \u03b5\u03c0\u03b1\u03bd\u03b1\u03bb\u03b1\u03bc\u03b2\u03b1\u03bd\u03cc\u03bc\u03b5\u03bd\u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7, \u03bc\u03c0\u03bf\u03c1\u03b5\u03af \u03bd\u03b1 \u03c7\u03c9\u03c1\u03b9\u03c3\u03c4\u03b5\u03af \u03c3\u03b5 \u03ad\u03bd\u03b1\u03bd \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc \u03b7\u03bc\u03b9\u03c4\u03bf\u03bd\u03bf\u03b5\u03b9\u03b4\u03ce\u03bd \u03ba\u03c5\u03bc\u03ac\u03c4\u03c9\u03bd \u03b4\u03b9\u03b1\u03c6\u03cc\u03c1\u03c9\u03bd \u03c3\u03c5\u03c7\u03bd\u03bf\u03c4\u03ae\u03c4\u03c9\u03bd \u03ba\u03b1\u03b9 \u03c0\u03bb\u03ac\u03c4\u03bf\u03c5\u03c2 (\u03b5\u03bd\u03c4\u03ac\u03c3\u03b5\u03c9\u03bd). \u0391\u03c5\u03c4\u03cc \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03bf \u03b1\u03c0\u03bf\u03c4\u03ad\u03bb\u03b5\u03c3\u03bc\u03b1 \u03c4\u03b7\u03c2 \u03b4\u03bf\u03c5\u03bb\u03b5\u03b9\u03ac\u03c2 \u03c0\u03bf\u03c5 \u03be\u03b5\u03ba\u03af\u03bd\u03b7\u03c3\u03b5 \u03bc\u03b5 \u03c4\u03bf\u03bd \u0393\u03ac\u03bb\u03bb\u03bf \u03bc\u03b1\u03b8\u03b7\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc Joseph Fourier, \u03bf \u03bf\u03c0\u03bf\u03af\u03bf\u03c2 \u03ad\u03b6\u03b7\u03c3\u03b5 \u03c4\u03b7 \u03b3\u03b1\u03bb\u03bb\u03b9\u03ba\u03ae \u03b5\u03c0\u03b1\u03bd\u03ac\u03c3\u03c4\u03b1\u03c3\u03b7 \u03c4\u03bf\u03bd \u03b4\u03ad\u03ba\u03b1\u03c4\u03bf \u03cc\u03b3\u03b4\u03bf\u03bf \u03b1\u03b9\u03ce\u03bd\u03b1. \u0397 \u03ad\u03ba\u03c6\u03c1\u03b1\u03c3\u03b7 \u03b5\u03bd\u03cc\u03c2 \u03b7\u03c7\u03b7\u03c4\u03b9\u03ba\u03bf\u03cd \u03ba\u03cd\u03bc\u03b1\u03c4\u03bf\u03c2 \u03ae \u03bf\u03c0\u03bf\u03b9\u03bf\u03c5\u03b4\u03ae\u03c0\u03bf\u03c4\u03b5 \u03c3\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2 \u03bc\u03b5\u03c4\u03b1\u03b2\u03ac\u03bb\u03bb\u03b5\u03c4\u03b1\u03b9 \u03bc\u03b5 \u03c4\u03b7\u03bd \u03c0\u03ac\u03c1\u03bf\u03b4\u03bf \u03c4\u03bf\u03c5 \u03c7\u03c1\u03cc\u03bd\u03bf\u03c5, \u03c9\u03c2 \u03c4\u03bf \u03ac\u03b8\u03c1\u03bf\u03b9\u03c3\u03bc\u03b1 \u03c4\u03c9\u03bd \u03b7\u03bc\u03b9\u03c4\u03bf\u03bd\u03bf\u03b5\u03b9\u03b4\u03ce\u03bd \u03ba\u03c5\u03bc\u03ac\u03c4\u03c9\u03bd \u03c0\u03bf\u03c5 \u03c4\u03bf \u03b1\u03c0\u03bf\u03c4\u03b5\u03bb\u03bf\u03cd\u03bd, \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03c9\u03c3\u03c4\u03ae \u03c9\u03c2\u00a0<em>\u03bc\u03b5\u03c4\u03b1\u03c3\u03c7\u03b7\u03bc\u03b1\u03c4\u03b9\u03c3\u03bc\u03cc\u03c2 Fourier<\/em>\u00a0\u03b1\u03c5\u03c4\u03bf\u03cd \u03c4\u03bf\u03c5 \u03c3\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2. (\u039c\u03c0\u03bf\u03c1\u03b5\u03af\u03c4\u03b5 \u03bd\u03b1 \u03b4\u03b9\u03b1\u03b2\u03ac\u03c3\u03b5\u03c4\u03b5 \u03bc\u03b9\u03b1 \u03c0\u03b9\u03bf \u03bb\u03b5\u03c0\u03c4\u03bf\u03bc\u03b5\u03c1\u03ae \u03b5\u03be\u03ae\u03b3\u03b7\u03c3\u03b7 \u03c4\u03c9\u03bd \u03bc\u03b1\u03b8\u03b7\u03bc\u03b1\u03c4\u03b9\u03ba\u03ce\u03bd \u03c0\u03bf\u03c5 \u03b5\u03bc\u03c0\u03bb\u03ad\u03ba\u03bf\u03bd\u03c4\u03b1\u03b9\u00a0<a href=\"https:\/\/plus.maths.org\/content\/os\/issue47\/features\/budd\/maths\" target=\"blank\" rel=\"noopener\">\u03b5\u03b4\u03ce<\/a>\u00a0\u2014 \u03c4\u03b1 \u03bc\u03b1\u03b8\u03b7\u03bc\u03b1\u03c4\u03b9\u03ba\u03ac \u03b5\u03af\u03bd\u03b1\u03b9 \u03b1\u03c1\u03ba\u03b5\u03c4\u03ac \u03c0\u03b5\u03c1\u03af\u03c0\u03bb\u03bf\u03ba\u03b1, \u03b1\u03bb\u03bb\u03ac \u03bf\u03b9 \u03bc\u03b1\u03b8\u03b7\u03bc\u03b1\u03c4\u03b9\u03ba\u03ad\u03c2 \u03b9\u03b4\u03ad\u03b5\u03c2 \u03c0\u03bf\u03c5 \u03b5\u03bc\u03c0\u03bb\u03ad\u03ba\u03bf\u03bd\u03c4\u03b1\u03b9 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c5\u03c0\u03ad\u03c1\u03bf\u03c7\u03b5\u03c2!)<\/p>\n<div class=\"centreimage\">\n<p><img decoding=\"async\" src=\"https:\/\/i0.wp.com\/plus.maths.org\/content\/sites\/plus.maths.org\/files\/articles\/2017\/carola\/Fourier_transform.gif?w=700&#038;ssl=1\" data-recalc-dims=\"1\" \/>\u0397 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7\u00a0<em>f<\/em>\u00a0\u03c0\u03bf\u03b9\u03ba\u03af\u03bb\u03bb\u03b5\u03b9 \u03c7\u03c1\u03bf\u03bd\u03b9\u03ba\u03ac \u2013 \u03b1\u03bd\u03c4\u03b9\u03c0\u03c1\u03bf\u03c3\u03c9\u03c0\u03b5\u03cd\u03b5\u03b9 \u03ad\u03bd\u03b1 \u03b7\u03c7\u03b7\u03c4\u03b9\u03ba\u03cc \u03ba\u03cd\u03bc\u03b1. \u0397 \u03b4\u03b9\u03b1\u03b4\u03b9\u03ba\u03b1\u03c3\u03af\u03b1 \u03bc\u03b5\u03c4\u03b1\u03c3\u03c7\u03b7\u03bc\u03b1\u03c4\u03b9\u03c3\u03bc\u03bf\u03cd Fourier \u03c0\u03b1\u03af\u03c1\u03bd\u03b5\u03b9 \u03c4\u03bf\u00a0<em>f<\/em>\u00a0\u03ba\u03b1\u03b9 \u03c4\u03bf \u03b1\u03c0\u03bf\u03c3\u03c5\u03bd\u03b8\u03ad\u03c4\u03b5\u03b9 \u03c3\u03c4\u03b1 \u03b7\u03bc\u03b9\u03c4\u03bf\u03bd\u03bf\u03b5\u03b9\u03b4\u03ae \u03ba\u03cd\u03bc\u03b1\u03c4\u03b1 \u03c0\u03bf\u03c5 \u03c4\u03b7\u03bd \u03b1\u03c0\u03bf\u03c4\u03b5\u03bb\u03bf\u03cd\u03bd, \u03bc\u03b5 \u03c3\u03c5\u03b3\u03ba\u03b5\u03ba\u03c1\u03b9\u03bc\u03ad\u03bd\u03b5\u03c2 \u03c3\u03c5\u03c7\u03bd\u03cc\u03c4\u03b7\u03c4\u03b5\u03c2 \u03ba\u03b1\u03b9 \u03c0\u03bb\u03ac\u03c4\u03b7. \u039f \u03bc\u03b5\u03c4\u03b1\u03c3\u03c7\u03b7\u03bc\u03b1\u03c4\u03b9\u03c3\u03bc\u03cc\u03c2 Fourier \u03b1\u03bd\u03b1\u03c0\u03b1\u03c1\u03af\u03c3\u03c4\u03b1\u03c4\u03b1\u03b9 \u03c9\u03c2 \u03b1\u03b9\u03c7\u03bc\u03ad\u03c2 \u03c3\u03c4\u03bf\u03bd \u03c4\u03bf\u03bc\u03ad\u03b1 \u03c3\u03c5\u03c7\u03bd\u03cc\u03c4\u03b7\u03c4\u03b1\u03c2, \u03bc\u03b5 \u03c4\u03bf \u03cd\u03c8\u03bf\u03c2 \u03c4\u03b7\u03c2 \u03b1\u03ba\u03af\u03b4\u03b1\u03c2 \u03bd\u03b1 \u03b4\u03b5\u03af\u03c7\u03bd\u03b5\u03b9 \u03c4\u03bf \u03c0\u03bb\u03ac\u03c4\u03bf\u03c2 \u03c4\u03bf\u03c5 \u03ba\u03cd\u03bc\u03b1\u03c4\u03bf\u03c2 \u03b1\u03c5\u03c4\u03ae\u03c2 \u03c4\u03b7\u03c2 \u03c3\u03c5\u03c7\u03bd\u03cc\u03c4\u03b7\u03c4\u03b1\u03c2.<\/p>\n<\/div>\n<hr \/>\n<h3>\u03a3\u03c7\u03b5\u03c4\u03b9\u03ba\u03ac \u03bc\u03b5 \u03b1\u03c5\u03c4\u03cc \u03c4\u03bf \u03ac\u03c1\u03b8\u03c1\u03bf<\/h3>\n<p>\u0391\u03c5\u03c4\u03cc \u03c4\u03bf \u03ac\u03c1\u03b8\u03c1\u03bf \u03b4\u03b7\u03bc\u03b9\u03bf\u03c5\u03c1\u03b3\u03ae\u03b8\u03b7\u03ba\u03b5 \u03c9\u03c2 \u03bc\u03ad\u03c1\u03bf\u03c2 \u03c4\u03b7\u03c2 \u03ba\u03ac\u03bb\u03c5\u03c8\u03ae\u03c2 \u03bc\u03b1\u03c2 \u03b3\u03b9\u03b1 \u03c4\u03b7\u03bd\u00a0<a href=\"https:\/\/www.newton.ac.uk\/event\/hyd2\/\" target=\"blank\" rel=\"noopener\"><em>\u03a5\u03b4\u03c1\u03bf\u03b4\u03c5\u03bd\u03b1\u03bc\u03b9\u03ba\u03ae \u0394\u03b9\u03b1\u03c3\u03c0\u03bf\u03c1\u03ac\u03c2: \u03bc\u03b1\u03b8\u03b7\u03bc\u03b1\u03c4\u03b9\u03ba\u03ac, \u03c0\u03c1\u03bf\u03c3\u03bf\u03bc\u03bf\u03af\u03c9\u03c3\u03b7 \u03ba\u03b1\u03b9 \u03c0\u03b5\u03b9\u03c1\u03ac\u03bc\u03b1\u03c4\u03b1, \u03bc\u03b5 \u03b5\u03c6\u03b1\u03c1\u03bc\u03bf\u03b3\u03ad\u03c2 \u03c3\u03b5 \u03c0\u03c1\u03cc\u03b3\u03c1\u03b1\u03bc\u03bc\u03b1 \u03bc\u03b7 \u03b3\u03c1\u03b1\u03bc\u03bc\u03b9\u03ba\u03ce\u03bd \u03ba\u03c5\u03bc\u03ac\u03c4\u03c9\u03bd<\/em><\/a>\u00a0\u03c0\u03bf\u03c5 \u03c6\u03b9\u03bb\u03bf\u03be\u03b5\u03bd\u03b5\u03af\u03c4\u03b1\u03b9 \u03b1\u03c0\u03cc \u03c4\u03bf \u0399\u03bd\u03c3\u03c4\u03b9\u03c4\u03bf\u03cd\u03c4\u03bf \u039c\u03b1\u03b8\u03b7\u03bc\u03b1\u03c4\u03b9\u03ba\u03ce\u03bd \u0395\u03c0\u03b9\u03c3\u03c4\u03b7\u03bc\u03ce\u03bd Isaac Newton. \u039c\u03c0\u03bf\u03c1\u03b5\u03af\u03c4\u03b5 \u03bd\u03b1 \u03b2\u03c1\u03b5\u03af\u03c4\u03b5 \u03c0\u03b5\u03c1\u03b9\u03c3\u03c3\u03cc\u03c4\u03b5\u03c1\u03bf \u03c0\u03b5\u03c1\u03b9\u03b5\u03c7\u03cc\u03bc\u03b5\u03bd\u03bf \u03b3\u03b9\u03b1 \u03c4\u03bf \u03c0\u03c1\u03cc\u03b3\u03c1\u03b1\u03bc\u03bc\u03b1\u00a0<a href=\"https:\/\/plus.maths.org\/content\/rainbows-rogue-waves-0\" target=\"blank\" rel=\"noopener\">\u03b5\u03b4\u03ce<\/a>\u00a0.<\/p>\n<p><a href=\"https:\/\/plus.maths.org\/content\/people\/index.html#rachel\" target=\"blank\" rel=\"noopener\">\u0397 Rachel Thomas<\/a>\u00a0\u03ba\u03b1\u03b9 \u03b7\u00a0<a href=\"https:\/\/plus.maths.org\/content\/people\/index.html#marianne\" target=\"blank\" rel=\"noopener\">Marianne Freiberger<\/a>\u00a0\u03b5\u03af\u03bd\u03b1\u03b9 \u03bf\u03b9 \u03c3\u03c5\u03bd\u03c4\u03ac\u03ba\u03c4\u03b5\u03c2 \u03c4\u03bf\u03c5\u00a0<em>Plus<\/em>\u00a0.<\/p>\n<p>\u0391\u03c5\u03c4\u03cc \u03c4\u03bf \u03ac\u03c1\u03b8\u03c1\u03bf \u03b2\u03b1\u03c3\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf \u03b2\u03b9\u03b2\u03bb\u03af\u03bf\u00a0<a href=\"http:\/\/www.amazon.co.uk\/Numericon-Journey-through-Hidden-Numbers\/dp\/1782061541\" target=\"blank\" rel=\"noopener\"><em>Numericon: \u0388\u03bd\u03b1 \u03c4\u03b1\u03be\u03af\u03b4\u03b9 \u03bc\u03ad\u03c3\u03b1 \u03c3\u03c4\u03b9\u03c2 \u03ba\u03c1\u03c5\u03c6\u03ad\u03c2 \u03b6\u03c9\u03ad\u03c2 \u03c4\u03c9\u03bd \u03b1\u03c1\u03b9\u03b8\u03bc\u03ce\u03bd<\/em><\/a>\u00a0\u03c4\u03c9\u03bd Marianne Freiberger \u03ba\u03b1\u03b9 Rachel Thomas \u03ba\u03b1\u03b9 \u03c3\u03c4\u03bf \u03ac\u03c1\u03b8\u03c1\u03bf \u03bc\u03b1\u03c2\u00a0<a href=\"https:\/\/plus.maths.org\/content\/fourier-transforms-images\" target=\"blank\" rel=\"noopener\"><em>Fourier transforms of images<\/em>\u00a0.<\/a><\/p>\n<p>&nbsp;<\/p>\n<p>https:\/\/plus.maths.org\/content\/why-sine-and-cosine-make-waves<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":2,"featured_media":55919,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"footnotes":"","_jetpack_memberships_contains_paid_content":false,"jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false}}},"categories":[16,33],"tags":[],"jetpack_publicize_connections":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v22.5 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ 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