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

Question

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

sin y = - 2/3, cos x = -1/5, x in quadrant II, y in quadrant III

Accepted Answer

Hey, everyone in this problem, we're asked to find the exact value of cosine of X minus Y. We're told to use a trigonometric identity and we're given some information. So we're told that sine of Y is equal to negative three divided by seven cosine of X is equal to negative two divided by nine X is in the second quadrant and Y is in the third quadrant. Now we have four answer choices here. Option A negative three multiplied by the square root of 77 plus four, multiplied by the square root of 10, all divided by 63. Option B negative four multiplied by the square root of 57 divided by 63. Option C is the negative of option A and option D is the negative of option B. OK. So that gives us those four answers. Now, we're gonna start with what we were given. So the expression we're interested in is cosine of X minus Y and we wanna use a trigonometric identity. So let's go ahead and recall the trigonometric identity for the difference when we're dealing with cosine. And that's gonna tell us that cosine of X minus Y is equal to cosine of X multiplied by cosine of Y, what sign of X multiplied by sign of why? Right. So this gives us a, us a way to break that expression. We had into some parts that may be easier to calculate or might be easier to find the values of given the information we have. Now, we already have sign of what and we already have cosine of X. We were given those in the problem. So what we need to do is figure out cosine of Y and sign of X. And if we can find those two values and we'll have everything we need to simplify the right hand side of our expression. All right. So let's start by finding sign of X. Now we're looking for a sin of X. So the angle we're interested in is the angle X. So we're gonna take what we know about cosine of X. OK? That it's equal to negative two divided by nine. And but the angle is in quadrant too. All right. Now, our call when we have cosine of an angle, right, cosine of an angle is going to be the horizontal or X component divided by R the radius. OK. So we can draw this out and we can write out these values. And because we're in quadrant two, that means that the X value, the horizontal component is going to be negative. OK. So that negative kind of goes with the numerator in this case, we can draw this out. And again, our angle is gonna be in quadrant too. And we know that it's going to have an X or horizontal component of two when we're dealing with just the magnitude. OK. And we're looking at coordinates, is it going to be negative two? And the ra radius or the hypotenuse of our triangle is going to be nine. OK. And we have our angle X. Now, what we're interested in finding is sin of X. OK. And recall that sin of X is going to be related and we're gonna call the vertical side side A and side of X is going to be equal to that side A divided by the hypotenuse of our triangle or the radius R. Um if we're taking it in coordinates like that. So sign of X in this case is going to be equal to a divided by nine. Now, what is a, well, we can find it? OK. This is a right angle triangle. We know two sides. So we can use the quadratic, sorry, not the quadratic equation. We can use py daggering theorem. OK. Right hand right angle triangle, we know two sides, we're gonna calculate the other. So we have that the hypotenuse squared nine squared is gonna be equal to the sum of the squares of the other two sides. So a squared plus two squared, simplifying 81 is equal to A squared plus four. That means that A squared is going to be equal to 77. And then we take the square root to get that A is equal to plus or minus the square root of 77. Now, this is really important when you're dealing with trig problems to remember that when you take the square root, you get both the positive and the negative root. OK. Now we have to infer from the problem which one to take. Now we're in quadrant two. So our Y values are positive. OK? So sign is going to be positive in quadrant two. So we are going to take the positive value. OK? Y is gonna be greater than zero. So we are gonna take A to be the square root of 77. OK? If A is the square root of 77 then our sign volume sine of X is going to be the square root of 77 divided by nine. Right. Now, we're gonna put a red box around that. We don't wanna lose track of that. We're gonna need that and we're gonna come back to it in just a minute. So we found one of our missing values sin of X. Now we're gonna do the same for cosine and Y hi. Now for cosine of Y what information are we given about angle? Y? Well, we're told that sine of Y is equal to negative 3/7 and that Y is in quadrant three. All right. So let's draw this out if Y is in quadrant three, that means the bottom left quadrant. OK. We have a negative Y value because we're below the X axis. So the negative in our sign is associated with that numerator. They recall that sign is the vertical or Y component divided by the radius. And so we can draw out our triangle just like we did for X. OK. So we have our angle Y we know that the vertical component as a magnitude of three, the radius or the hypotenuse is seven. And we are gonna call this missing horizontal side B. All right. Now, how do we find beat? Well, we know we know sine of Y already, we want to find cosine of Y and we know that cosine of Y is going to be equal to that side B divided by our hypotenuse or that side B divided by seven. OK. So we are interested in finding B and again, we can do it through Pythagorean theorem because we have a right hand hand triangle and we know two of the sides already. So hypotenuse squared, seven squared is equal to the sum of the squares of the other two sides, three squared plus B squared. And this tells us at 49 is equal to nine plus B squared. Subtracting nine on both sides, 40 is equal to B squared. And finally taking the square root B is equal to plus or minus the square root of 40. Now we are in quadrant three and in quadrant three, we're to the left of the Y axis which tells us that our X values are negative. All right. In other words, cosign is negative. Hm What this tells us is that our B value is going to be negative the square root of 40. All right. So we can write then that cosine of Y is going to be equal to negative the square root of 40 divided by seven. All right. So we're getting there, let's remind ourselves of what we're trying to do. We are trying to find cosine of X minus Y. We use our difference equation, we wrote out this expression and in order to solve, we needed cosine of Y and sin of X which we now have. So we can go to this equation now and substitute in the values we know. So cosine of X negative two nights cosine of Y, we just found to be negative square root of 40 divided by seven. OK. Sign of X, we just found to be the square root of 77 divided by nine. A sign of why we were given in the problem as negative three sevens. OK? We can go ahead and simplify in the first term, we have a negative multiplied by a negative that's gonna give us a positive. So we get two multiplied by the square root of 40 all divided by 63. The second term we have a positive multiplied by negative. So we're gonna get a negative. So we subtract and we have three multiplied by the square root of 77. Again divided by 63. We have this common denominator. Now, we can simplify the square root of 40 K 40 can be written as four multiplied by 10. And we know that the square root of four is equal to two. OK. So we can simplify this and write it as four multiplied by the square root of 10. And we've taken the four outside the square of that is 22, multiplied by two, gives us that four out in front minus three multiplied by the square of 77. And this is all divided by 63. And we have a common denominator here. So we can simplify this into one fraction. And that is the final value that is the exact value of cosine of X minus Y that we were looking for. If we can compare this to our answer choices, we can see that option A is correct. Our answer is the same as negative three, multiplied by the square root of 77 plus four, multiplied by the square root of 10, divided by 63. Thanks everyone for watching. I hope this video helped you in the next one.

Use the given information to find cos(x - y).
sin y = - 2/3, cos x = -1/5, x in quadrant II, y in quadrant III

Key Concepts

Trigonometric Identities

Quadrants and Sign of Trigonometric Functions

Finding Missing Trigonometric Values

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