Use a half-angle identity to find each exact value.
sin ...

Question

Use a half-angle identity to find each exact value.

sin 165°

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the given trigonometric expression using the half angle identities. We're given the sign of 67.5 degrees. Our answer choices are a, the square root of two minus the square root of two divided by two B, the square root of under the square root, we have two plus the square root of two. All of which is divided by four, see the square root of two plus the square root of two under the radical, all of which is divided by two D, the square root of two minus the square to two under the radical, all of which divided by four. So first, since we are told we want to use a half angle identity, I'm gonna write that the half angle identity for sign is we have the sine of angle A divided by two equals plus or minus the square root of one minus the cosine of a, all of which divided by two. So first, we need to decide what is a, well, we know we're using a half angle identity and that we have the sine of a divided by two has equaled the sign of 67.5 degrees. So this means a divided by two equals 67.5. I'm gonna multiply both sides by two to get the A by itself. And we find that our angle here has to be A, is 135 degrees. Now that we know that A is 135 degrees. I'm going to use that into this identity. So we will get that the sign of 67.5 degrees equals and it says plus or minus, but 67.5 is in quadrant one. So this is only gonna be positive. So the square root of one minus the cosine of 135 degrees divided by two. So using a unit circle, I'm going to find what is the cosine of 135 degrees and plug that in. So I get the square root of one minus and then the cosine of 135 degrees is negative radical two divided by two, all of which is divided by two. I'm going to simplify the numerator under my radical. So I'm gonna treat one as if it's two divided by two and combine it with the negative radical two divided by two. And when I do that, I get two plus the square to two divided by two, all of which is again divided by two under my square root symbol. Thinking about this vertically I'm gonna rewrite this under the radical still as two plus the square root of two over two. I'm sorry, divided by two. And then since that whole thing was divided by two, again, I'm gonna multiply it by the reciprocal of two, which is one half. And this will give me the square root of two plus the square root of two divided by four. Simplifying my square roots, I'm gonna take the square root of both the numerator and the denominator. So I have the square root of two plus the square to two over the square root of four, the square root of four is two. So this means in the end, the sign of 67.5 degrees is equal to the square root of two plus the square to two divided by two. So that is our answer and that matches answer choice C have a nice day.

Use a half-angle identity to find each exact value.
sin 165°

Key Concepts

Half-Angle Identities

Reference Angles

Exact Values of Trigonometric Functions

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