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

sin s = 2/3 and sin t = -1/3, s in quadrant II and t in quadrant IV​​

Video transcript

Hello, today we're going to be determining the exact values of cosine of X plus Y and cosine of X minus Y. Before we jump into the problem, let's quickly write down the trigger to metric identity for both of these values cosine of X plus Y is given to us as cosine of X multiplied by cosine of Y minus sine of X multiplied by sine of Y. Now, the trigonometric identity for cosine of X minus Y is similar to that of X plus Y. The only difference being a sine change. So cosine of X minus Y is given to us as cosine of X multiplied by cosine of Y and instead of minus, it'll be plus sine of X multiplied by sine of Y. So we want to find the two values using the trigonometric identities. The problem tells us that sin of X is equal to three divided by 13 and sine of Y is equal to negative two divided by seven. But in order to get the two values, we need the values of cosine of X and cosine of Y. So what we're going to need to do is we'll first need to solve for cosine of X and cosine of Y. Let's start by solving for cosine of X. In order to solve for cosine of X, we'll be using the Pythagorean identity sine squared of X plus cosine squared of X is equal to one. Let's go ahead and solve for cosine in the given equation. If we subtract sine squared to both sides of the equation that will leave us with cosine squared of X is equal to one minus sine squared of X. If we were to take the square root of both sides of the equation, that will leave us with cosine of X is equal to the positive or negative square root of one minus sine squared of X. Now, before we plug in any values, we need to determine whether we want to keep the positive or negative value for cosine of X. Well, the problem tells us that the angle X exists in quadrant, one of the unit circle in quadrant, one of the unit circle cosine is a positive value. So we will be keeping the positive value for cosine of X. And since we know that the value of sin of X is three divided by 13 plugging in that value will allow us to rewrite the equation as cosine of X is equal to the positive square root of one minus the quantity three divided by 13 squared. Algebraically simplifying the inside of the square root will leave us with the square root of 160 divided by 169. And further algebraically simplifying the square root will leave us with the value of four multiplied by the square root of 10 divided by 13. This will be the value for cosine of X. Now, let's go ahead and solve for cosine of Y. Now, just like cosine of X, we'll be using the same pythagorean identity. However, we'll be using it with respect to the Y angle. So we'll be using sine squared of Y plus cosine squared to Y is equal to one. Now, just like before, if we were to solve for cosine of Y, we will be left with cosine of Y is equal to the positive or negative square root of one minus sine squared of Y. Now, again, we need to determine whether we want to keep the positive or negative value of the square root. Well, in the beginning of the problem, we are told that the angle Y exists within the third quadrant of the unit circle in the third quadrant of the unit circle cosine is a negative value. So we'll be keeping the negative value of the square root. Furthermore, we do know that sine of Y is equal to negative two divided by seven. So if we were to plug in these values, we can rewrite cosine of Y as cosine of Y is equal to the negative square root of one minus the quantity negative two divided by seven squared. Algebraically simplifying the inside of the square root will leave us with the negative square root of 45 divided by 49. And further simplifying the square root will leave us with the final value of negative three multiplied by the square root of five divided by seven. This is going to be the value of cosine of Y. Now that we have both cosine of X and cosine of Y, we can plug it into the trigonometric identity for cosine of X plus Y and cosine of X minus Y. That way we may be able to solve for those values. So let's go ahead and plug in our values to cosine of X plus Y. Now recall cosine of X plus Y is equal to cosine of X which we solve to be four multiplied by the square root of 10 divided by 13 multiplied by cosine of Y which we solved to be negative three multiplied by the square root of five divided by seven minus sine of X, which was given to us as three divided by 13 multiplied by sin of Y, which was given to us as negative two divided by seven. Multiplying the left two fractions will give us the fraction negative 60 multiplied by the square root of two divided by 91. And multiplying the two right fractions will give us the value of negative six divided by 91 negative one multiplied by negative six divided by 91 will allow us to rewrite the equation as negative 60 multiplied by the square root of two divided by 91 plus six, divided by 91. And if we were to reorder the terms and combine the fractions with respect to the positive number, our final value for cosine of X plus Y will give us six minus 60 multiplied by the square root of two divided by 91. This will be the value for cosine of X plus Y. Now we're going to repeat this process for the value of cosine of X minus Y. Now, the setup is going to be similar to that of X plus Y. The only difference is that we have a positive sign instead of a negative sign. So X minus Y will equal to four multiplied by the square root of 10 divided by 13 multiplied by negative three, multiplied by the square root of five, divided by seven plus three, divided by 13 multiplied by negative two, divided by seven. The simplification will be similar to that of cosine of X plus Y. We will have negative 60 multiplied by the square root of two divided by 91. But now we have plus the quantity negative six divided by 91 combining the two fractions together will give us the value negative 60 multiplied by the square root of two minus six divided by 91. Furthermore, if we reorder the terms of the numerator and factor out a negative one, we can rewrite the final value to be negative six plus 60 multiplied by the square root of two divided by 91. And this will be the value for cosine of X minus Y. So just to recap the final value of cosine of X plus Y is six minus 60 multiplied by the square root of two divided by 91. And the co the value of cosine X minus Y is negative six plus 60 multiplied by the square root of two divided by 91. With that being said, the answer to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.