{"id":41,"date":"2011-12-28T00:27:59","date_gmt":"2011-12-28T00:27:59","guid":{"rendered":"https:\/\/schooltutoring.com\/help\/?p=41"},"modified":"2014-12-02T08:33:51","modified_gmt":"2014-12-02T08:33:51","slug":"trigonometry-definition-of-a-radian","status":"publish","type":"post","link":"https:\/\/schooltutoring.com\/help\/trigonometry-definition-of-a-radian\/","title":{"rendered":"Trigonometry &#8211; Definition of a Radian"},"content":{"rendered":"<p>We&#8217;ve seen how to measure angles in degrees using a protractor:<\/p>\n<p><a href=\"https:\/\/schooltutoring.com\/wp-content\/uploads\/sites\/2\/protractor.png\"><img loading=\"lazy\" decoding=\"async\" title=\"protractor\" src=\"https:\/\/schooltutoring.com\/wp-content\/uploads\/sites\/2\/protractor-300x202.png\" alt=\"\" width=\"300\" height=\"202\" \/><\/a><\/p>\n<p>However, the degree is a fairly arbitrary unit of measurement \u2013 who decided to divide the circle into 360 parts? Why can\u2019t the circle be divided into 100 parts? 1000 parts?<\/p>\n<p><strong>Radian<\/strong> measure is an alternative way to measure angles. Instead of dividing the circle into an arbitrary number of parts, we look at the length of the arc that subtends the angle. We measure the angle based on this length as a ratio to the radius.<\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"https:\/\/schooltutoring.com\/wp-content\/uploads\/sites\/2\/Radian1.png\"><img loading=\"lazy\" decoding=\"async\" title=\"Radian\" src=\"https:\/\/schooltutoring.com\/wp-content\/uploads\/sites\/2\/Radian1-300x238.png\" alt=\"\" width=\"300\" height=\"238\" \/><\/a><\/p>\n<p>One radian is defined as the angle where the length of the arc equals the length of the radius.\u00a0If we traverse the circle completely, we\u2019ll have travelled the length of the circumference.<\/p>\n<p><em>C = 2\u03c0r<\/em><\/p>\n<p>Using this, we can find the number of radians in a circle. The length of the circumference, as a ratio to the radius, is:<\/p>\n<p>&nbsp;<\/p>\n<p><em># of radians in a circle = arc length \/radius<\/em><\/p>\n<p>Applying this to a circle, if we rearrange this equation to isolate radians, we can find the number of radians in a given number of degrees:<\/p>\n<p><em>radians =\u00a0360 degrees \/\u00a02\u03c0<\/em><\/p>\n<p><em>= 180 \/ \u03c0 degrees<\/em><\/p>\n<p>Rearrange to obtain:<\/p>\n<p><em>degrees = \u03c0\/180 radians<\/em><\/p>\n<p>Using this formula, we can find the number of degrees in a given number of radians. To summarize, radian measure is an alternative way to measure angles. Instead of dividing the circle into an arbitrary number of parts, we look at the length of the arc that subtends the angle. We measure the angle based on this length as a ratio to the radius.<\/p>\n<p>We can convert between radian and degree measure using the relationship that 2\u03c0 radians equals 360 degrees; they both measure the angles in a circle:<\/p>\n<p><em>2\u03c0 radians=360 degrees<\/em><\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>We&#8217;ve seen how to measure angles in degrees using a protractor: However, the degree is a fairly arbitrary unit of measurement \u2013 who decided to divide the circle into 360 parts? Why can\u2019t the circle be divided into 100 parts? 1000 parts? Radian measure is an alternative way to measure angles. Instead of dividing the [&hellip;]<\/p>\n","protected":false},"author":19,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"inline_featured_image":false,"footnotes":""},"categories":[18],"tags":[],"class_list":["post-41","post","type-post","status-publish","format-standard","hentry","category-trigonometry"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/41","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/users\/19"}],"replies":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/comments?post=41"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/41\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/media?parent=41"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/categories?post=41"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/tags?post=41"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}