{"id":236126,"date":"2025-06-16T05:08:18","date_gmt":"2025-06-16T05:08:18","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=236126"},"modified":"2025-06-16T05:08:21","modified_gmt":"2025-06-16T05:08:21","slug":"if-sin-theta-1-2-then-find-the-value-of-theta","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/16\/if-sin-theta-1-2-then-find-the-value-of-theta\/","title":{"rendered":"If sin theta = 1 \/2 then find the value of theta ."},"content":{"rendered":"\n

If sin theta = 1 \/2 then find the value of theta .<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Given:sin\u2061\u03b8=12\\sin \\theta = \\frac{1}{2}sin\u03b8=21\u200b<\/p>\n\n\n\n

Correct Answer:<\/strong>\u03b8=30\u2218or\u03b8=150\u2218\\theta = 30^\\circ \\quad \\text{or} \\quad \\theta = 150^\\circ\u03b8=30\u2218or\u03b8=150\u2218<\/p>\n\n\n\n

(or in radians: \u03b8=\u03c06or\u03b8=5\u03c06\\theta = \\frac{\\pi}{6} \\quad \\text{or} \\quad \\theta = \\frac{5\\pi}{6}\u03b8=6\u03c0\u200bor\u03b8=65\u03c0\u200b)<\/p>\n\n\n\n

\n\n\n\n

Explanation<\/strong><\/p>\n\n\n\n

The sine function relates an angle in a right-angled triangle to the ratio of the length of the side opposite the angle to the hypotenuse. The value of sine is positive in both the first and second quadrants of the unit circle. To find the angle when sin\u2061\u03b8=12\\sin \\theta = \\frac{1}{2}sin\u03b8=21\u200b, consider the standard angles used in trigonometry.<\/p>\n\n\n\n

Looking at the unit circle, the sine of 30\u221830^\\circ30\u2218 is known to be exactly 12\\frac{1}{2}21\u200b. This is derived from the properties of a 30-60-90 triangle, where the side ratios are 1 (opposite 30\u00b0), 3\\sqrt{3}3\u200b (opposite 60\u00b0), and 2 (the hypotenuse). Therefore:sin\u206130\u2218=12\\sin 30^\\circ = \\frac{1}{2}sin30\u2218=21\u200b<\/p>\n\n\n\n

However, sine is also positive in the second quadrant. The angle in the second quadrant that has the same sine value is found using the identity:sin\u2061(180\u2218\u2212\u03b8)=sin\u2061\u03b8\\sin(180^\\circ – \\theta) = \\sin \\thetasin(180\u2218\u2212\u03b8)=sin\u03b8<\/p>\n\n\n\n

Thus:sin\u2061150\u2218=sin\u2061(180\u2218\u221230\u2218)=12\\sin 150^\\circ = \\sin(180^\\circ – 30^\\circ) = \\frac{1}{2}sin150\u2218=sin(180\u2218\u221230\u2218)=21\u200b<\/p>\n\n\n\n

So, both 30\u221830^\\circ30\u2218 and 150\u2218150^\\circ150\u2218 satisfy the equation. These are the principal values in the interval 0\u22180^\\circ0\u2218 to 360\u2218360^\\circ360\u2218.<\/p>\n\n\n\n

In radian measure, these angles convert to:30\u2218=\u03c06,150\u2218=5\u03c0630^\\circ = \\frac{\\pi}{6}, \\quad 150^\\circ = \\frac{5\\pi}{6}30\u2218=6\u03c0\u200b,150\u2218=65\u03c0\u200b<\/p>\n\n\n\n

Because the sine function is periodic with a period of 360\u2218360^\\circ360\u2218 (or 2\u03c02\\pi2\u03c0 radians), other possible solutions can be expressed as:\u03b8=30\u2218+360\u2218nor\u03b8=150\u2218+360\u2218n\\theta = 30^\\circ + 360^\\circ n \\quad \\text{or} \\quad \\theta = 150^\\circ + 360^\\circ n\u03b8=30\u2218+360\u2218nor\u03b8=150\u2218+360\u2218n<\/p>\n\n\n\n

for any integer nnn, representing the full set of angles where sin\u2061\u03b8=12\\sin \\theta = \\frac{1}{2}sin\u03b8=21\u200b.<\/p>\n\n\n\n

\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

If sin theta = 1 \/2 then find the value of theta . The Correct Answer and Explanation is: Given:sin\u2061\u03b8=12\\sin \\theta = \\frac{1}{2}sin\u03b8=21\u200b Correct Answer:\u03b8=30\u2218or\u03b8=150\u2218\\theta = 30^\\circ \\quad \\text{or} \\quad \\theta = 150^\\circ\u03b8=30\u2218or\u03b8=150\u2218 (or in radians: \u03b8=\u03c06or\u03b8=5\u03c06\\theta = \\frac{\\pi}{6} \\quad \\text{or} \\quad \\theta = \\frac{5\\pi}{6}\u03b8=6\u03c0\u200bor\u03b8=65\u03c0\u200b) Explanation The sine function relates an angle in a right-angled […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-236126","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/236126","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=236126"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/236126\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=236126"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=236126"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=236126"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}