Match each expression in Column I with its value in Column II....

Question

Match each expression in Column I with its value in Column II.

10. cos 67.5°

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the cosine of 112.5 degrees. Our answer choices are a negative square root under the square root symbol. We have two plus the square to two. All of which is divided by two B. We have a negative square root of under the square root symbol, we have two minus the square root of two, all of which is divided by two C, we have negative and then a square root symbol under the square root, we have two minus the square root of two, all of which is divided by four D, the square root and under the square root, we have two minus the square root of two, all of which is divided by two. So first, this looks like a good candidate for our half angle identities. So the cosine of an angle A divided by two is plus or minus the square root of one plus the cosine of angle A divided by two. So in this case, we need to find out what is A. So we have the cosine of 112.5 degrees. We're going to say that's equal to the cosine of angle A divided by two. So we could say a divided by two is equal to 112.5. And if I double that by multiplying both sides by two, to isolate the A I get the angle A for this example, must be 225 degrees. So using that into our identity, we're gonna say that the cosine of 225 degrees divided by two equals plus or minus the square root of one plus the cosine of 225 degrees. All of which is divided by two. Since our original angle, 112.5 degrees is in quadrant two, we can say that our answer here will be a negative. So I'm gonna make it negative now and work with it through. So this will be a negative square root of one plus I'm gonna use a unit circle to find what is the cosine of 225 degrees. And it appears to be a negative square root of two divided by two. And all of that is now also divided by two. So combining what's in my numerator, I still have a negative and then a square root one is the same as two divided by two. So I can rewrite this as two minus the square root of two divided by two again, divided by two. I'm going to rewrite this fraction so that I have a complex fraction where both the numerators fraction and the denominators have a denominator of two. So I have this negative square root of two minus the square root of two divided by two, all of which will be divided by four, divided by two. Since they have the same denominator, they will be able to cancel out. So I have negative square root of two minus the square root of two divided by four. Simplifying this, I'm gonna take the square root of both the numerator and the denominator. So I have the negative square root of two minus the square to two divided by the square root of four. Since the square root of four is two, I could say this is negative square root of two minus the square root of two in the numerator divided by two. So that should be your answer that the cosine of 112.5 degrees is equal to the negative square root of two minus the square to two, all of which is divided by two. And that is answer choice B have a nice day.

Match each expression in Column I with its value in Column II.
10. cos 67.5°

Key Concepts

Cosine of Special Angles

Half-Angle Formula

Sign Determination in Trigonometric Values

Watch next