In Exercises 57–64, find the exact value of the following unde...

Question

In Exercises 57–64, find the exact value of the following under the given conditions:

a. cos (α + β)

cos α = 8/17, α lies in quadrant IV, and sin β = -1/2, β lies in quadrant III.

Accepted Answer

Hello, today we're going to be determining the value of cosine of X plus Y using the following conditions. So we can expand cosine of X plus Y by using the sum property of cosine and the sum property of cosine states that if we have cosine of X plus Y, this statement can be expanded as cosine of X multiplied by cosine of Y minus sine of X multiplied by sine of Y. So in order to find the value of cosine of X plus Y, we're going to need to solve for our cosine of X sine X values and our cosine of Y and sine Y values. So how do we go about doing that? Well, we can solve for cosine and sine of X using the first condition and we can solve for cosine and sine of Y using our second condition. Let's take a look at the first condition. We are told that cosine of X is equal to 21/29. And this angle of X exists in the fourth quadrant of the unit circle. So one thing to note is that this is going to be our cosine of X value but we'll still need to use this value to solve for sin of X. And one more thing to note is that since this angle exists in quadruple four, quadruple four represents any X value as positive and any Y value as negative. So how can we use this this angle to solve for code sine of X while we can create a right triangle that exists in quadruple four? Now recall that cosine for any angle is defined as the length of the triangle over its hypotenuse. So we can use cosine of X equals to 21/29 to label the length of the triangle as 21 its hypo news as 29. But now we need to use this right triangle to solve for sin of X. Recall that sign of any right triangle is defined as the height of the triangle over its hypotenuse. How do we solve for the height of the triangle? Well, we can use the Pythagorean theorem and the Pythagorean theorem states that the hypotony squared is equal to the length of the triangle squared plus its height squared. The hypotenuse is 29 the X value is gonna be 21. So if we plug this into the Python theorem, we get 29 squared is equal to 21 squared plus Y squared. 29 squared is going going to give us the value of 841 and 21 squared gives us the value of 441. So tracking 441 to both sides is going to give us 400 is equal to Y squared. And if we take the square root of both sides of the equation, we get Y is equal to 20. So the height of this right triangle is going to be 20. But keep in mind that since this triangle is in quadrant four, any Y value or the height of the triangle is going to be negative. So the height of this triangle is going to be negative 20. And now if we use the identity for sin of X, we get sign of X is equal to the height which is negative over the hypotenuse which is 29. So we have our cosine of X and sin of X values. Now we need to solve for our cosine of Y and sine of Y values. Let's take a look at the second condition. The second condition states that sign of Y is equal to negative 1/7. And the angle of Y lies in the third quadrant of the unit circle. Since the angle of Y is in the third quadrant, the third quadrant represents any X value as negative and any Y value as negative. And just like before we can use sine of Y to solve for cosine of Y. So let's create a right triangle in the third quadrant. Now again, sign for any right triangle is defined as the height of the triangle over its hypotenuse. So the height of the triangle is going to be negative one and its hypo is going to be seven. So what we need to do now is we need to solve for the length of the triangle. In order to get cosine of Y, we're going to use the Pythagorean theorem one more time in order to solve for the length of the triangle. In this case here, the hypotenuse is going to be seven and the height of Y is going to be negative one. So plugging this into the Pythagorean theorem is going to give us seven squared is equal to X squared plus negative one squared, seven squared gives us the value of 49 negative one squared gives us positive one. So tracking one on both sides leaves us with X squared is equal to 48. And taking the square root of both sides of this equation is going to give us an X value of four square root of three. So the length of this triangle is going to be four square root of three. But recall that any X value or the length of the triangle is going to be negative in quadrant three. So the length of this triangle is going to be negative four square root of three. And that means cosine of Y is going to equal to negative four square root of 3/7. So just to summarize, we solved for our cosine of Y value, our sign of X value and we were given cosine of X and sign of Y, we can take these values and plug it into the addition property for cosine in order to solve for cosine of X plus Y. So we stated that cosine of X is equal to 21/29. So we have 21/29 multiplied by cosine of Y which is negative four square root of 3/7. And we're going to be subtracting sine of X which was given to us as negative 20/ multiplied by sign of Y which is given to us as negative 1/7. Now, we need to multiply the values together 21/29 multiplied by negative four square root of 3/ is going to give us the value of negative 84 square root of three, over 203 and negative 20/29 multiplied by negative 1/7 is going to give us the value of positive 20 over 203. But keep in mind that we are still subtracting this value. So we're going to get a negative value, we can then combine these two values together into a single fraction and we are left with negative 84 square root of three minus 20 over 203. We can factor out a negative sign or a negative one from the numerator. And that is going to leave us with negative one multiplied by the quantity 84 square root of three plus 20 over 203. And this is going to be the exact value of cosine of X plus Y. And with that being said, the answer to this problem is going to be D. 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.

In Exercises 57–64, find the exact value of the following under the given conditions:
a. cos (α + β)
cos α = 8/17, α lies in quadrant IV, and sin β = -1/2, β lies in quadrant III.

Key Concepts

Sum of Angles Formula for Cosine

Determining Sine and Cosine Values from Quadrants

Using Pythagorean Identity to Find Missing Values

Watch next