Find the exact value of each expression. (Do not use a calculator.)

cos π/12

Video transcript

Hello, everyone. We are asked to determine the exact value of the following trigonometric function without using a calculator. We want to find the value of the cosine of seven pi divided by 12. Our answer choices are a, the square root of six minus the square root of two, all of which is divided by four B, the square root of six plus the square root of two, all of which is divided by four C, the square to two minus the square to six, all of which is divided by four D negative square root of two minus the square root of six, all of which is divided by four. So looking at what we are given, we are given again the cosine of seven pi divided by 12. I think the easiest way to find the exact value here is to rewrite this as a sum of two angles. So seven pi divided by 12, I can break this down into the cosine of four pi divided by 12 plus three pi divided by 12. And then I can reduce that to be the cosine of pi divided by three plus pi divided by four. So that's what I'm gonna work with remembering are some formula or identity for cosine. It is the cosine of X plus Y equals the cosine of X multiplied by the cosine of Y minus the sine of X multiplied by the sine of Y. So for our purposes, I'm saying X is pi divided by three. Why is pie divided by four? So the cosine of pi divided by three plus pi divided by four will equal the cosine of pi divided by three, multiplied by the cosine of pi divided by four minus the sine of pi divided by three, multiplied by the sine of pi divided by four to find the values of the cosine and sine here, I'm gonna use a unit circle. So the cosine of pi divided by three on my unit circle has a value of one half. The cosine of pi divided by four has a value of the square root of two divided by two minus. The sine of pi divided by three is the square root of three divided by two. And the sine of pi divided by four is also the square root of two divided by two. So now let's multiply our fractions together. So one half multiplied by the square to two divided by two will give us the square to two divided by four minus. And then the square root of three divided by two multiplied by the square of two, divided by two will be the square root of six divided by four. Since these have a common denominator, I can rewrite this over a common denominator of four. And my numerator will be the square to two minus the square to six. So our answer for the exact value here is the square to two minus the square root of six, all of which is divided by four. And that's answer choice C have a nice day.