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.

cos² 105° - sin² 105°

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 cosine squared of 112.5 degrees minus the sine squared of 112.5 degrees. Our answer choices are A, the squared of two divided by two B negative one half C negative radical, three divided by two D negative radical two divided by two. First, looking at my cosine double angle identities, I recognize that the one that would work best in this situation would be that the cosine of two theta equals the cosine squared of theta minus the sine squared of theta. I think this one works best because if I match it up with what we are given as our original expression, which was cosine squared of 112. degrees minus sine squared of 112.5 degrees. We're in a very similar format and we can say that in this case, theta would be 112.5 degrees. So now if I take that value of theta and plug it into the part where we have that it's the cosine of two theta, we will have the cosine of two multiplied by 112.5 degrees. This would be the cosine of 225 degrees. When I reference a unit circle, I find that the cosine of 225 degrees is negative radical two divided by two. So using the cosine of a double angle, we found that the exact value is negative radical two divided by two which is answer choice D have a nice day.

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. cos² 105° - sin² 105°

Key Concepts

Double Angle Formulas

Pythagorean Identity

Exact Values of Trigonometric Functions

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