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 105° (Hint: 105° = 60° + 45°)

Accepted Answer

Welcome back. Everyone. In this problem, we want to determine the exact value of the cosine of 165 degrees without using a calculator. For our answer choices. A says is the square of six minus the square root of two all divided by four. B says the square of six plus the square of two all divided by four. C says the square of two minus the square root of six, all divided by four. And the D says a negative square root of two minus the square root of six all divided by four. Now, in this problem, we want the exact value for this cosine function, but just by looking at it, we wouldn't know the exact value of the cosine of 165 degrees right off the bat. However, we could rewrite 165 degrees as a sum and recall that the sum identity for the cosine tells us that the cosine of a sum X plus Y is going to be equal to the cosine of X multiplied by the cosine of Y minus the sine of X multiplied by the sine of Y. So in other words, this trigonometric identity helps us to break it down in an easier format. Now, if we could rewrite the cosine of 165 degrees as the values of cosines and signs that we know, then we would be able to find its exact value. How can we do that? Well, if we think about it, we could rewrite the cosine of 165 degrees as the cosine of 120 degrees plus 45 degrees. It's more manageable for us to figure out what those exact values are. So since let me write, since here, since the cosine of 165 degrees could be written as that sum, then applying the sum identity, the cosine of 120 degrees plus 45 degrees is going to be equal to the cosine of 120 degrees multiplied by the cosine of 45 degrees minus the sine of 120 degrees multiplied by the sine of 45 degrees. So now if we can figure out those four expressions, their exact values, then we will be able to solve for the cosine of 165 degrees. Now, how can we do that? Well, let's see if we can use the unit circle to help us. No recall. Let's say that we have our unit circuit here. Uh X and Y and for our unit circle, recall that each coordinate on our unit circle can be written in or is written in the form cosine theta, sine theta where cosine theta is the X value and sine theta is the Y value. No. How I like to think about it. I like to plot some of my angles here that are greater than 90 degrees on the unit circle and then try to match them up with the one that I know the ones in the first quadrant, the ones that would reside somewhere there. So let's put the cosine first, let's find 100 and 20 degrees on our unit circle, which should be somewhere right here. OK. And we need to find a cosine of 120 degrees and the cosine of and, and the sign of 120 degrees. Likewise here, we can also find our 45 degree angle. OK? 45 degrees. Well, I think 45 degrees would be a little bit more up here. I, and we know that we want to find a cosine and sine of our 45 degree angle and our 120 degree angle. Now, how are we going to do that? Well, one way that I like to think about it. OK. If, if, if we think about it here, let's start with our 120 degree angle. Notice that it's really 60 degrees away from 180 degrees. OK. So it's also going to be, if I were to mirror it in the first quadrant here. Then it would mirror kind of mirror what's going on with a 60 degree angle. OK. Now, when we think about it here, right, what do we know about our 60 degree Angus? Well, let's start with the cosine. The cosine is the X value. So for a 60 degree angle, in other words, we can say that the cosine of 120 degrees is going to be equal to the negative value of the cosine of 60 degrees. Because both of them correspond here in the quadrants. It's just that they are mirror of each other. So if we know what the cosine of 60 is, then we can find a cosine of 120. Well, we know that a cosine of 60 degrees is a half. So the cosine of 120 degrees is going to be negative a half. Now, if we think about it here, if we're thinking about the sine of 120 degrees, the sine of 120 degrees is the Y value. And if we come across, we can realize that it shares the same value as the sine of 60 degrees. So the sine of 120 degrees equals the sine of 60 degrees. And the sine of 60 degrees is the square root of three divided by two. So now we have the cosine and sine of 120 degrees. How about their 45 degree angles? Well, the cosine of 45 degrees is equal to the sine of 45 degrees. And both of them are equal to the square root of two divided by two. If we were to find them on our unit circle, those would have been the values that we found. And if we're not sure about why we found out with our unit circle, we could also use our special triangle to help us. But we have no our values for the cosine and sine of 45 degrees and 120 degrees. So that means no, we can find a cosine of 165 degrees because no, this tells us then that a cosigner our son, OK is going to be equal to a cosine of 120 degrees, which is negative a half multiplied by the cosine of 40 degrees, which is the square root of two divided by two minus the sine of 120 degrees, which is the square of three divided by two multiplied by the sine of 45 degrees. The square of two divided by two. Now we can go ahead and simplify negative one, multiplied by the square of two equals negative square of two and two multiplied by two equals four. The spirit of three multiplied by the square of two equals the square of 62 multiplied by two equals four. Since both of them share the same denominator, we can rewrite the numerators or both numerators in the same. So that's the negative square of two minus the spirit of six all divided by four. So that is what our sum equal to. And thus the cosine of 165 degrees. If we look back on our answer choices, that would have been answer choice. Thanks a lot for watching everyone. I hope this video helped.

Find the exact value of each expression. (Do not use a calculator.)
cos 105° (Hint: 105° = 60° + 45°)

Key Concepts

Angle Sum Identity for Cosine

Exact Values of Common Angles

Simplification of Radicals

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