In Exercises 39–42, use double- and half-angle formulas to find t...

Question

In Exercises 39–42, use double- and half-angle formulas to find the exact value of each expression.sin 22.5°

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the expression using the half ale formula. We are given the sine of 15 degrees. We are also given four answer choices. So using a half angle formula, I'm gonna use the sign of theta divided by two equals plus or minus the square root of one minus the cosine of theta divided by two. Here we are given the sine of 15 degrees, 15 degrees is half of 30 degrees. So if we say that theta for our example, so theta divided by two, this would be the sine of 30 degrees divided by two. So we're gonna say theta equals 30 degrees. This helps us because now we can find the cosine of theta being the cosine of 30 degrees using a unit circle and find that it is radical three divided by two. I can plug that into my half angle formula. And since 15 degrees is in quadrant one, this will be positive. So it will be the square root of one minus the square root of three divided by two divided by two. I'm going to simplify this if I can. And I think the easiest way to do that would be to first write my numerator as a single fraction. So I will have the square root of two minus radical three divided by two, all of which is divided by two. And then I will rewrite this as the square root of two minus radical three divided by two and then divided by 2/1 using my fraction rules, I'm going to rewrite this one more time as multiplication. So I will have the square root of two minus radical three divided by two, multiplied by one half. And that will give me the square root of two minus radical three divided by four. If I break this down a little bit, I will have this numerator as the square root of two minus radical three. Uh The denominator would be the square of four which is two. So I would have the square root of two minus radical three divided by two. And I'm not sure that this can get any simpler. No, that looks like all we can do here. So our answer is the square root of two minus radical three, all of which is divided by two which will be answer choice. A have a nice day.

In Exercises 39–42, use double- and half-angle formulas to find the exact value of each expression.sin 22.5°

Key Concepts

Double-Angle Formulas

Half-Angle Formulas

Exact Values of Trigonometric Functions

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