Use the given information to cos(x - y).
cos x = 2/9, si...

Question

Use the given information to cos(x - y).

cos x = 2/9, sin y = -1/2, x in quadrant IV, y in quadrant III

Accepted Answer

Hello, everyone. We are asked to find the exact value of the cosine of X minus Y. We will use trigonometric identities. We are given that the cosine of X equals 5/11 and the sine of Y equals negative 1/8 X is in the fourth quadrant, Y is in the third quadrant. Our answer choices are a negative 15 radical seven divided by 71 B negative 15 radical seven plus four radical six divided by 88 C 15 radical seven divided by 71 D 15 radical seven minus four radical six, all of which is divided by 88. The first thing I'm going to do is write my identity for the cosine of X minus Y and that is the cosine of X cosine of Y plus the sine of X sine of Y. So noticing that I am given the cosine of X and the sine of Y, I need to go find the sine of X still and the cosine of Y luckily through manipulating our Pythagorean identities, I can say that the sine of X equals plus or minus the square root of one minus the cosine squared of X since X is in quadrant four. I know that sine will have to be negative. So we are gonna be dealing with the negative root. So we'll have negative radical one minus our cosine of X which is 5/11 squared. So the sin of X equals negative radical one minus 25 121st or 25 divided by 121 rewriting this. So the one under the radical has a common denominator will have the sine of X equals negative one. I'm sorry, equals negative radical. 121 divided by 121 minus 25 divided by 121. So this would be the negative square root of we'll have 96 divided by 121 under a radical. So the sin of X is gonna be negative square root of 96 divided by 121. And if I take the square root of both the numerator and the denominator, I get negative square root of 96 divided by 11, the square root of 96 can be simplified. So we'll have the negative and then it will be four radical six divided by 11. So now we know what the sine of X is. Let's find the cosine of Y. And lucky for us, a similar identity comes from the Pythagorean identities. So I can say that the cosine of Y equals plus or minus the square root of one minus the sine squared of Y. So Y is in the third quadrant, which means cosine will be negative. So the cosine of Y will equal negative square root of one minus our sine of Y. So negative 1/8 squared, the cosine of Y will equal negative square root of one minus 1 64th, rewriting the one as a common denominator, I'll have negative square root of 64 64th minus 1 64th. So the cosine of Y will equal negative and it'll be the square root of 63 64th which if I take the square root of both the numerator and the denominator, the negative is gonna stay out front. I have the square of 63 divided by eight and I am pretty sure I can reduce the square to 63 to 3 radical seven. So I'll have the cosine of Y equals negative three radical seven divided by eight. All right. So now that I know all the pieces I can use my a cosine of X minus Y identity. So the cosine of X minus Y is going to equal my cosine of X which is 5/11 multiplied by the cosine of Y which we just found to be negative three radical seven divided by eight plus the sine of X which we found to be negative four radical six divided by 11 multiplied by the sign of Y which we were given negative 1/8. So let's multiply our fractions, the cosine of X minus Y will be negative 15 radical seven divided by 88 plus. My next set of fractions I'll end up with a positive four radical six divided by 88. And when I combine this into one fraction over a common denominator, I get negative 15 radical seven plus four radical six, all of which is divided by 88. And I don't believe that can be simplified any further. So this is our answer and it matches answer choice. B negative 15 radical seven plus four radical six, all of which is divided by 88. Have a nice day.

Use the given information to cos(x - y).
cos x = 2/9, sin y = -1/2, x in quadrant IV, y in quadrant III

Key Concepts

Trigonometric Identities - Cosine of a Difference

Determining Signs of Trigonometric Functions by Quadrant

Using Pythagorean Identity to Find Missing Values

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