{"id":46487,"date":"2025-07-02T06:50:38","date_gmt":"2025-07-02T06:50:38","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=46487"},"modified":"2025-07-02T06:50:39","modified_gmt":"2025-07-02T06:50:39","slug":"find-the-exact-value-of-arcsin","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/find-the-exact-value-of-arcsin\/","title":{"rendered":"Find the exact value of arcsin"},"content":{"rendered":"\n

Find the exact value of arcsin\\left(-\\frac{\\sqrt{3}}{2}\\right). a.) \\text{ }arcsin\\left(-\\frac{\\sqrt{3}}{2}\\right) = -\\frac{\\pi}{3} b.) \\text{ }arcsin\\left(-\\frac{\\sqrt{3}}{2}\\right) = \\frac{4\\pi}{3} c.) \\text{ }arcsin\\left(-\\frac{\\sqrt{3}}{2}\\right) = -\\frac{\\pi}{6} d.) \\text{ }arcsin\\left(-\\frac{\\sqrt{3}}{2}\\right) = \\frac{\\pi}{3}<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

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

The correct answer is a.) arcsin(-\u221a3\/2) = -\u03c0\/3<\/strong>.<\/p>\n\n\n\n

Here is a detailed explanation:<\/p>\n\n\n\n

The expression arcsin(-\u221a3\/2) asks for the angle, let’s call it \u03b8, whose sine is equal to -\u221a3\/2. This can be written as the equation sin(\u03b8) = -\u221a3\/2.<\/p>\n\n\n\n

A critical aspect of the inverse sine function, arcsin(x), is its restricted range. To ensure that the function gives a single, unique output for each input, its range is defined as [-\u03c0\/2, \u03c0\/2]. In degrees, this is equivalent to [-90\u00b0, 90\u00b0]. This means our final answer must be an angle in the first or fourth quadrant, with fourth quadrant angles represented as negative values.<\/p>\n\n\n\n

To find the value of \u03b8, we can first identify the reference angle. We do this by considering the positive version of the value: sin(\u03b8) = \u221a3\/2. From our knowledge of the unit circle and standard trigonometric values, we know that the angle in the first quadrant whose sine is \u221a3\/2 is \u03c0\/3 (or 60\u00b0). This is our reference angle.<\/p>\n\n\n\n

Now, we must determine the correct quadrant for our original problem, sin(\u03b8) = -\u221a3\/2. The sine function is negative in the third and fourth quadrants. However, we must choose the angle that falls within the [-\u03c0\/2, \u03c0\/2] range of the arcsin function. The third quadrant is outside this range. The fourth quadrant is included. Using our reference angle of \u03c0\/3, the angle in the fourth quadrant is -\u03c0\/3.<\/p>\n\n\n\n

This angle, -\u03c0\/3, is within the required range of [-\u03c0\/2, \u03c0\/2]. We can verify our answer by checking if sin(-\u03c0\/3) equals -\u221a3\/2. Since sine is an odd function, sin(-x) = -sin(x). Therefore, sin(-\u03c0\/3) = -sin(\u03c0\/3) = -\u221a3\/2. This confirms our result.<\/p>\n\n\n\n

Comparing this with the given options, option (a) is -\u03c0\/3. Option (b), 4\u03c0\/3, is incorrect because while sin(4\u03c0\/3) is indeed -\u221a3\/2, this angle is not in the principal value range of the arcsin function. Options (c) and (d) are incorrect as they correspond to different sine values.thumb_upthumb_down<\/p>\n\n\n\n

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

Find the exact value of arcsin\\left(-\\frac{\\sqrt{3}}{2}\\right). a.) \\text{ }arcsin\\left(-\\frac{\\sqrt{3}}{2}\\right) = -\\frac{\\pi}{3} b.) \\text{ }arcsin\\left(-\\frac{\\sqrt{3}}{2}\\right) = \\frac{4\\pi}{3} c.) \\text{ }arcsin\\left(-\\frac{\\sqrt{3}}{2}\\right) = -\\frac{\\pi}{6} d.) \\text{ }arcsin\\left(-\\frac{\\sqrt{3}}{2}\\right) = \\frac{\\pi}{3} The Correct Answer and Explanation is: The correct answer is a.) arcsin(-\u221a3\/2) = -\u03c0\/3. Here is a detailed explanation: The expression arcsin(-\u221a3\/2) asks for the angle, let’s call it \u03b8, whose sine […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-46487","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/46487","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=46487"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/46487\/revisions"}],"predecessor-version":[{"id":46491,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/46487\/revisions\/46491"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=46487"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=46487"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=46487"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}