{"id":38903,"date":"2025-06-26T14:45:56","date_gmt":"2025-06-26T14:45:56","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=38903"},"modified":"2025-06-26T14:45:57","modified_gmt":"2025-06-26T14:45:57","slug":"find-the-exact-value-cos-135","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/find-the-exact-value-cos-135\/","title":{"rendered":"Find the exact value. cos 135\u00b0"},"content":{"rendered":"\n

Find the exact value. cos 135\u00b0<\/p>\n\n\n\n

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

The exact value of cos 135\u00b0<\/strong> is: \u221222\\boxed{-\\dfrac{\\sqrt{2}}{2}}\u221222\u200b\u200b\u200b<\/p>\n\n\n\n

Explanation <\/h3>\n\n\n\n

To find the exact value of cos 135\u00b0<\/strong>, we first recognize that this angle lies in the second quadrant of the unit circle. In the second quadrant, cosine values are negative<\/strong> because cosine corresponds to the x-coordinate<\/strong> of a point on the unit circle, and x-values are negative in that region.<\/p>\n\n\n\n

Next, we use the concept of a reference angle<\/strong>. A reference angle is the acute angle formed between the terminal side of a given angle and the x-axis. For 135\u00b0, its reference angle is: 180\u00b0\u2212135\u00b0=45\u00b0180\u00b0 – 135\u00b0 = 45\u00b0180\u00b0\u2212135\u00b0=45\u00b0<\/p>\n\n\n\n

This means that the cosine of 135\u00b0 has the same magnitude as the cosine of 45\u00b0, but with a negative sign since it is in the second quadrant.<\/p>\n\n\n\n

From trigonometric values of special angles, we know: cos\u206145\u00b0=22\\cos 45\u00b0 = \\dfrac{\\sqrt{2}}{2}cos45\u00b0=22\u200b\u200b<\/p>\n\n\n\n

Since 135\u00b0 is in the second quadrant, we apply the negative sign: cos\u2061135\u00b0=\u2212cos\u206145\u00b0=\u221222\\cos 135\u00b0 = -\\cos 45\u00b0 = -\\dfrac{\\sqrt{2}}{2}cos135\u00b0=\u2212cos45\u00b0=\u221222\u200b\u200b<\/p>\n\n\n\n

This value is exact because it comes from a well-known special angle (45\u00b0), which is part of the 30\u00b0-60\u00b0-90\u00b0<\/strong> and 45\u00b0-45\u00b0-90\u00b0<\/strong> right triangle relationships on the unit circle.<\/p>\n\n\n\n

This value can also be understood geometrically. On the unit circle, a 135\u00b0 angle corresponds to a point in the second quadrant that forms a 45\u00b0 angle with the negative x-axis. The coordinates of this point are: (\u221222,22)\\left(-\\dfrac{\\sqrt{2}}{2}, \\dfrac{\\sqrt{2}}{2}\\right)(\u221222\u200b\u200b,22\u200b\u200b)<\/p>\n\n\n\n

Since cosine is the x-coordinate, this confirms that: cos\u2061135\u00b0=\u221222\\cos 135\u00b0 = -\\dfrac{\\sqrt{2}}{2}cos135\u00b0=\u221222\u200b\u200b<\/p>\n\n\n\n

In summary, cosine values depend on both the angle\u2019s reference and the quadrant in which it lies. Understanding the unit circle and reference angles is key to finding exact trigonometric values without a calculator.<\/p>\n\n\n\n

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

Find the exact value. cos 135\u00b0 The Correct Answer and Explanation is: The exact value of cos 135\u00b0 is: \u221222\\boxed{-\\dfrac{\\sqrt{2}}{2}}\u221222\u200b\u200b\u200b Explanation To find the exact value of cos 135\u00b0, we first recognize that this angle lies in the second quadrant of the unit circle. In the second quadrant, cosine values are negative because cosine corresponds […]<\/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-38903","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/38903","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=38903"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/38903\/revisions"}],"predecessor-version":[{"id":38905,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/38903\/revisions\/38905"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=38903"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=38903"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=38903"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}