Find cos(s + t) and cos(s - t).

cos s = √2/4 and sin t = - √5/6, s and t in quadrant IV

Video transcript

Hello, everyone. We are asked to determine the exact value of the cosine of X plus Y and the cosine of X minus Y. The cosine of X is negative radical five divided by eight. The sine of Y is negative radical seven divided by four X and Y are both in the third quadrant. Our answer choices are a, the cosine of X plus Y equals three radical 59 plus the square root of 35. All of which is divided by 32. The cosine of X minus Y equals three radical 59 minus the square to 35. All of which is divided by 32 B. The cosine of X plus Y equals three radical 59 minus the square of 35. All of which is divided by 32. The cosine of X minus Y equals three radical 59 plus the square root of 35 all of which is divided by 32 C. The cosine of X plus Y equals negative radical 413 plus three radical five, all of which is divided by 32. The cosine of X minus Y equals the square root of 413 plus three radical five, all of which is divided by 32 D. The cosine of X plus Y equals the square root of 413 plus three radical five. All of which is divided by 32. The cosine of X minus Y equals negative radical 413 plus three radical five, all of which is divided by 32. So to begin, I'm gonna write the identities needed to solve this. So the cosine of X plus Y has an identity. That is the cosine of X cosine of Y minus the sine of X sine of Y. And the cosine of X minus Y equals the cosine of X cosine of Y plus the sine of X sine of Y. So noticing in both of these identities, I need to find the cosine of Y and the sine of X. So I'm gonna start by trying to find the sign of X using a Pythagorean identity. I recall that the sine of X will equal the square root, sorry plus or minus the square root of one minus the cosine squared of X. Since we're in quadrant three, we only need to worry about the negative root. So we are going to use the sin of X equals negative square root of one minus our cosine of X just negative radical five divided by eight squared. So when I square this, I get the sin of X equals negative radical one minus and negative squared is a positive, the squared of five squared is five and eight squared is 64. I'm going to rewrite my one as 64 divided by 64. So I have negative square root of 64 divided by 64 minus five, divided by 64. And I get that the sine of X equals the negative square root of 59 divided by 64. Picking the square root of both the numerator and the denominator. I get that this is negative radical 59 divided by eight. So my sin of X is negative radical 59 divided by eight. So let's see, what else do I still need? I need to find the cosine of Y. So luckily the same Pythagorean identity helps us out here, we can state that the cosine of Y will equal plus or minus the square root of one minus the sine squared of Y. So let's find the cosine of Y. Again, it's gonna be a negative because we're in quadrant three. So negative square root of one minus and our sine of Y was negative radical seven divided by four squared. So our cosine of Y will equal negative square root of one minus and then a negative squared will be a positive. The square root of seven squared is seven and four squared is 16, rewriting this I have the negative square root of 16 divided by 16 minus seven divided by 16. So I get that the cosine of Y equals the negative square root of nine divided by 16. And when I take the square root of both the numerator and denominator, I get that this is negative 3/4. So the cosine of Y will be negative 3/4. All right. Now that we have all of our values, let's plug them in. I'm gonna draw a line just to differentiate my work. So I'm going to start with the cosine of X plus Y. The cosine of X, we have negative radical five divided by eight multiplied by the cosine of Y that we just found to be negative 3/4 minus the sine of X which we found to be negative radical 59 divided by eight multiplied by the sine of Y which we have is negative radical seven divided by four. All right, let's see what we get when I multiply straight across with my fractions. My first set of fractions, I get a positive three radical five divided by 32. And I'm gonna be subtracting and now multiplying negative radical 59 by negative radical seven. And that will give me a positive square root of 413 divided by 32 rewriting this as a single fraction. Um I will rewrite it. So I have my negative up front, I think and I'm going to write it as negative radical 413 plus three radical five divided by 32. And that can't be simplified any further. So that is the cosine of X plus Y. Now to find the cosine of X minus Y, we're gonna use the same numbers. The difference is between our multiplication will be a plus sign. So we'll still have negative radical five divided by eight multiplied by negative 3/4. But now we're adding that the negative radical 59 divided by eight multiplied by negative radical seven divided by four. Our multiplication comes out to be the same, which is very nice. So we have three radical five divided by 32 plus the square root of 413 divided by 32. And since I put the larger square root first, last time, I'm gonna do it again this time and we'll have the cosine of X minus Y equals the square root of 413 plus three radical five, all of which is divided by 32. And so now we have both parts of our answer. Let's look for a match. Um Let's see, quickly glancing. It's either C or D and then matching up our negative signs. It is answer choice C. The cosine of X plus Y is negative radical 413 plus three radical five, all of which is divided by 32 and the cosine of X minus Y is the square root of 413 plus three radical five, all of which is divided by 32 have a nice day.