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)

sin α = 3/5, α lies in quadrant I, and sin β = 5/13, β lies in quadrant II.

Accepted Answer

Hello, today we're going to be solving for the exact value of cosine of X plus Y using the following conditions. So we want to solve for cosine of X plus Y. In order to solve for the exact value, we can expand this quantity by using the sum identity for cosine the sum identity for cosine states that cosine of X plus Y is equal to cosine of X multiplied by cosine of Y minus sine of X multiplied by sine of Y. So in order to solve for cosine of X plus Y, we're going to need our cosine of X cosine of Y value and our sign of X and sign of Y value. So how do we go about obtaining these four values? Well, we're going to need to use the two conditions that are given to us. Let's take a look at the first condition. We are told that sign of X is equal to 8/17 and the angle of X lies in the first quadrant of the unit circle. Now there's a few things to note about this first because the quadrant is quadrant one, any X and Y value in quadrant one is going to be positive. And we can use the given sine value to make a right triangle in quadrant one in order to solve for cosine of X. So we are given our sign of X value, which is one of the values we need, let's create a right triangle that is in quadruple one. Now recall that sign of any right triangle is defined as the height of the triangle over its hypotenuse. Since we are given sign of X is equal to 8/17, the height of the triangle is going to be eight and its hypotenuse is going to be 17. And what we need to do is sell for cosine cosine of any right triangle is defined as the length of the triangle over its hypotenuse. But currently we are missing the length of the triangle we can solve for the length of the triangle by using the Pythagorean theorem. And the Pythagorean theorem states that the hypo squared is equal to X squared plus Y squared. The hypo news is going to be 17 and the height Y is going to be eight. And if we plug this into the equation, we get 17 squared is equal to X squared plus eight squared, 17 squared is going to evaluate to give us 289 and that is equal to X squared plus eight squared, which is going to give us 64. So tracking 64 to both sides of the equation is going to give us X squared is equal to 225. And taking the square root of both sides of this equation is going to give us X is equal to 15. So that tells us that the length of this right triangle is going to be 15. And using the identity for cosine of a right triangle, we get cosine of X is equal to 15/17. So now we have the cosine of X value. What, what, what is left is the sulfur sign of Y and cosine of Y? And we can do that by taking a look at the second condition. The second condition states that sign of Y is equal to 12/37. And we are told that the angle of Y exists in quadrant two. Now, since this angle is in quadrant two, any values of X and quadrant two is going to be negative and any values of Y are going to be positive. So just like before we're going to create a right triangle, but this right triangle is going to be in quadrant two. And using the identity for sign of X, we can label the height of the triangle as 12 and the hypotenuse as 37. So in order to solve for cosine, we're going to need to solve for the length of the triangle, we can do this by using the Pythagorean theorem one more time. Now, the hypotenuse of this triangle is going to be 37 its height is going to be 12. So plugging this into the equation gives us 37 squared is equal to X squared plus 12 squared. 37 squared, evaluates to 1369 and 12 squared evaluates to 144 subtracting 1 44 to both sides of the equation gives us X squared as equal to 1225. And taking the square root of both sides of the equation gives us X as equal to 35. So we can label the length of the triangle as 35. But keep in mind any length of the triangle or X value in quadrant two is going to be negative. So the length of this triangle is going to be negative 35. And using the identity for cosine, we get cosine of Y is equal to negative 35/12 over 37. So just to summarize, we have our sin of X, our sin of Y values and we have our cosine of X and cosine of Y values, we can plug this into the sum identity for cosine of X plus Y to solve for its value. So what we get is cosine of X which is equal to 15/17, multiplied by cosine of Y which is negative 35/37. And we're going to be subtracting sign of X which is 8/17, multiplied by sign of Y which is 12/ 15/17, multiplied by negative 35. Over 37 gives us negative over 629 and 8/ multiplied by 12. Over 37 gives us 96 over 629. Combining these fractions together gives us the value of negative 525 minus 96 over 629. And an enumerator subtracting the two values together is going to give us negative 621 over 629. 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 B 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)
sin α = 3/5, α lies in quadrant I, and sin β = 5/13, β lies in quadrant II.

Key Concepts

Sum of Angles Formula for Cosine

Determining Cosine from Sine and Quadrant

Quadrant Sign Rules for Trigonometric Functions

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