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

Question

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

cos (-7π/12)

Accepted Answer

Hello, everyone. We're asked to determine the exact value of the following trigonometric function. Without using a calculator. We are given the cosine of negative 11 pi divided by 12. Our answer choices are a, the squared to six minus the square to two. All of which is divided by four B negative square root of two minus the square root of six, all of which is divided by four C, the square to six minus the square to two, all of which is divided by two D negative square root of two minus the square root of six, all of which is divided by two. First, looking at our original function, the cosine of negative 11 pi divided by 12. I'm reminded of an identity for odd and even functions that stated that the cosine of negative X is equal to the cosine of X. So for our case, we could say that the cosine of negative 11 pi divided by 12 is going to be equal to the cosine of 11 pi divided by 12. And that makes this one step simpler from there without having a calculator. I think I need to break down 11 pi divided by 12 into a sum of two angles. So I can say 11 pi divided by 12 is equal to eight pi divided by 12 plus three pi divided by 12. And if I reduce those two fractions, I can say this is equivalent to two pi divided by three plus pi divided by four. These values help us cause they're on our unit circle. So that's what we wanna use. We wanna make sure we have values that are on our unit circle from there. Recall the identity for the sum of angles of a cosine is the cosign 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 example, X will be two pi divided by three and Y will be pi divided by four. So let's plug those values in. So we can say the cosine of two pi divided by three plus pi divided by four equals the cosine of two pi divided by three, multiplied by the cosine of pi divided by four minus the sign of two pi divided by three multiplied by the sine of pi divided by four. And at this point, I will use a unit circle to find the exact values. So the value of the cosine of two pi divided by three is negative one half. This will be multiplied by the cosine of pi divided by four, which is the square root of two divided by two. And then I'm subtracting the value for the sine of two pi divided by three, which is the square root of three divided by two multiplied by the sine of pi divided by four, which has a value of the square root of two divided by two. So now I'll do my multiplication of the fractions and then I'll subtract. So for my first set of fractions negative one multiplied by the square root of two is negative square root of two divided by four. My second group of fractions, the square root of three multiplied by the square root of two is the square root of 62, multiplied by two is four. They already have a common denominator. So we can rewrite this as one single fraction and we'll have negative square to two minus the square to six all which is divided by four. That is our answer. That is the exact value. And that's answer. Choice B have a nice day.

Find the exact value of each expression. (Do not use a calculator.)
cos (-7π/12)

Key Concepts

Unit Circle

Cosine Function

Angle Reference and Quadrants

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