{"id":7680,"date":"2022-09-16T07:16:04","date_gmt":"2022-09-16T07:16:04","guid":{"rendered":"https:\/\/mdr.foobrdigital.com\/?p=7680"},"modified":"2022-09-16T07:16:04","modified_gmt":"2022-09-16T07:16:04","slug":"unit-circle-and-trigonometric-values","status":"publish","type":"post","link":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/2022\/09\/16\/unit-circle-and-trigonometric-values\/","title":{"rendered":"Unit Circle and Trigonometric Values"},"content":{"rendered":"\n<p>The various trigonometric identities and their principal angle values can be calculated through the use of a unit circle. In the unit circle, we have cosine as the x-coordinate and sine as the y-coordinate. Let us now find their respective values for \u03b8 = 0\u00b0, and \u03b8 = 90\u00ba.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/mdr.foobrdigital.com\/wp-content\/uploads\/2022\/09\/gggg.png\" alt=\"\" class=\"wp-image-7681\"\/><\/figure>\n\n\n\n<p>For \u03b8 = 0\u00b0, the x-coordinate is 1 and the y-coordinate is 0. Therefore, we have cos0\u00ba = 1, and sin0\u00ba = 0. Let us look at another angle of 90\u00ba. Here the value of cos90\u00ba = 1, and sin90\u00ba = 1. Further, let us use this unit circle and find the important trigonometric function values of \u03b8 such as 30\u00ba, 45\u00ba, 60\u00ba. Also, we can also measure these \u03b8 values in\u00a0radians. We know that 360\u00b0 = 2\u03c0 radians. We can now convert the angular measures to radian measures and express them in terms of the radians.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Unit Circle Table:<\/h3>\n\n\n\n<p>The unit circle table is used to list the coordinates of the points on the unit circle that corresond to common angles with the help of trigonometric ratios.<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-regular\"><table><thead><tr><th scope=\"col\">Angle \u03b8<\/th><th scope=\"col\">Radians<\/th><th scope=\"col\">Sin\u03b8<\/th><th scope=\"col\">Cos\u03b8<\/th><th scope=\"col\">Tan\u03b8 = Sin\u03b8\/Cos\u03b8<\/th><th scope=\"col\">Coordinates<\/th><\/tr><\/thead><tbody><tr><td>0\u00b0<\/td><td>0<\/td><td>0<\/td><td>1<\/td><td>0<\/td><td>(1, 0)<\/td><\/tr><tr><td>30\u00b0<\/td><td>\u03c0\/6<\/td><td>1\/2<\/td><td>\u221a3\/2<\/td><td>1\/\u221a3<\/td><td>(\u221a3\/2, 1\/2)<\/td><\/tr><tr><td>45\u00b0<\/td><td>\u03c0\/4<\/td><td>1\/\u221a2<\/td><td>1\/\u221a2<\/td><td>1<\/td><td>(1\/\u221a2, 1\/\u221a2)<\/td><\/tr><tr><td>60\u00b0<\/td><td>\u03c0\/3<\/td><td>\u221a3\/2<\/td><td>1\/2<\/td><td>\u221a3<\/td><td>(1\/2, \u221a3\/2)<\/td><\/tr><tr><td>90\u00b0<\/td><td>\u03c0\/2<\/td><td>1<\/td><td>0<\/td><td>undefined<\/td><td>(0,1)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>We can find the secant, cosecant, and cotangent functions also using these formulas:<\/p>\n\n\n\n<ul><li>sec\u03b8 = 1\/cos\u03b8<\/li><li>cosec\u03b8 = 1\/sin\u03b8<\/li><li>cot\u03b8 = 1\/tan\u03b8<\/li><\/ul>\n\n\n\n<p>We have discussed the unit circle for the first quadrant. Similarly, we can extend and find the radians for all the unit circle quadrants. The numbers 1\/2, 1\/\u221a2, \u221a3\/2, 0, 1 repeat along with the sign in all 4 quadrants.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Unit Circle in Complex Plane<\/h2>\n\n\n\n<p>A unit circle consists of all\u00a0complex numbers\u00a0of absolute value as 1. Therefore, it has the equation of |z| = 1. Any complex number z = x +\u00a0iiy will lie on the unit circle with equation given as x<sup>2<\/sup>\u00a0+ y<sup>2<\/sup>\u00a0= 1.<\/p>\n\n\n\n<p>The unit circle can be considered as unit complex numbers in a complex plane, i.e., the set of complex numbers z given by the form,<\/p>\n\n\n\n<p>z = e<sup>iit<\/sup>&nbsp;= cos t +&nbsp;ii&nbsp;sin t = cis(t)<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The various trigonometric identities and their principal angle values can be calculated through the use of a unit circle. In the unit circle, we have cosine as the x-coordinate and sine as the y-coordinate. Let us now find their respective values for \u03b8 = 0\u00b0, and \u03b8 = 90\u00ba. For \u03b8 = 0\u00b0, the x-coordinate [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[453],"tags":[],"_links":{"self":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts\/7680"}],"collection":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/comments?post=7680"}],"version-history":[{"count":0,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts\/7680\/revisions"}],"wp:attachment":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/media?parent=7680"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/categories?post=7680"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/tags?post=7680"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}