In Exercises 15–22, write each expression as the sine, cosine, or...

Question

In Exercises 15–22, write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression.

𝝅
2 cos² ------ ﹣ 1
8

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the following expression by writing it as the cosine of a double angle. The expression we are given is two multiplied by the cosine squared of 11 pi divided by minus one. We have four answer choices. A radical three divided by two B negative radical two divided by two C negative one half D negative radical three divided by two. The directions say we should have the cosine of a double angle. So I'm gonna write down the identity of the cosine of a double angle. So I'm just using generically two theta. And that, that identity is that the cosine of two theta would equal two multiplied by the cosine squared of theta minus one. So this does look similar to what we are given. We are given two multiplied by the cosine squared of 11 pi divided by minus one. So this means when we match this up, then our example theta equals 11 pi divided by 12. So if I want to plug 11 pi divided by 12 in for theta in the double angle portion, the portion that says the cosine of two theta I can do so. So I'd have the cosine of two multiplied by 11 pi divided by 12. And when I simplify two, multiplied by 11 pi divided by 12, I will get the cosine of 11 pi divided by six which looking at a unit circle, I know that the value of that is radical three divided by two. So in the end, the exact value of the expression we are given equals radical three divided by two, which is answer choice. A have a nice day.

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In Exercises 15–22, write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. 𝝅 2 cos² ------ ﹣ 1 8