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)

tan α = 3/4, 𝝅 < α < 3𝝅/2, and cos β = 1/4, 3𝝅/2 < β < 2𝝅

Accepted Answer

Hello, today we're gonna be determining the exact value of the expression cosine of X plus Y. So we want to find the value of cosine of X plus Y cosine of X plus Y can be expanded by using the sum rule of cosine. And this value can be expanded as cosine of X multiplied by cosine of Y minus sine of X multiplied by sign of Y. In order to get this value, we're going to need our cosine of X and cosine of Y values and our sign of X and sin of Y values. But how can we get those values? Why we can use the given conditions to help us obtain those values? So let's take a look at condition one condition one tells us that tangent of X is equal to 20/21. And the angle of X lies between the region of pi and three pi over two. The quadrant that lies between pi and three pi over two on the unit circle is going to be quadrant three and any values in quadrant three are represented as negative X comma negative Y. So what we need to do is we need to use this condition to obtain cosine of X and sin of X. But how can we do that? Well, we can do that by drawing a right triangle that exists in quadrant three. Now recall that tangent for any right triangle is defined as the height of the triangle Y over the length of the triangle X. And since we are told that tangent of X is equal to 20/21 the height of the triangle is going to be 20 and the length of the triangle is gonna be 21. But again, recall that any length X and any height Y is going to be negative in quadrant three. So we can represent these side links of the triangle as negative 21 negative 20. So we need to get sign of X and cosine of X recall that sign of X is defined as the height of a triangle over its hypotenuse and cosine of X is defined as the length of the triangle over the hypotenuse. The only value we're missing from this right triangle is its hypotenuse. But we can solve for the hypo news by using the Python theorem. The Pythagorean theorem states that the hypotenuse squared is equal to X squared plus Y squared. In this case here X is negative 21 Y is negative 20. So we get the hypo squared is equal to the value of negative 21 squared plus negative 20 squared. Negative 21 squared gives us the value of positive and negative 20 squared gives the value of positive 400. Adding these values together is going to give us 841. And the hypotenuse squared is equal to 841. Finally taking the square root of both sides of this equation is going to give us the hypotenuse is equal to 29. So the hypotenuse for this right triangle is going to be 29. And if we use the identities for sign and cosine of a right triangle sine of X is going to equal to negative over 29 cosine of X is equal to negative 21/29. We have our cosine of X and cosine of uh cosine of X and sign of X value. But now we need to get our cosine of Y and S sign of Y values. We can do that by using the second condition. Now, the second condition states that cosine of Y is equal to 1/2. And the region that the angle wide lies in is going to be between three pi over two and two pie. The region that lies between three pi over two and two pi is going to be quadrant four of the unit circle and any X value in quadrant four is going to be positive, any Y value is going to be negative. So just like before we can use cosine of Y is equal to one half in order to solve for sine of Y. So we do need a cosine of Y value. So at least we know cosine of Y is equal to one half. Let's go ahead and create a right triangle that exists in quadrant four. Now again, cosine is defined as the length of the triangle overheads, pipo news. So the length of the triangle is going to be one and its hypo news is going to be two. Now we need to solve for the height of the triangle because sine of Y is going to be defined as the height of the triangle over hit IPO News. And we can do that by using the Pythagorean theorem one more time. So we have the hypotenuse squared is equal to the length X squared plus the height Y squared. But this time we're going to be solving for Y, the hypotenuse is going to be two and the length is gonna be one. So we get two squared is equal to one squared plus Y squared. Two squared is going to evaluate to give us four and one squared is going to evaluate to give us one. So tracking one of both sides of the equation is going to give us three is equal to Y squared. And taking the square root of both sides of this equation is going to give us Y is equal to the square root of three. So the height of the triangle is going to be the square root of three. But recall that any Y value or the height of the triangle is going to be negative in quadruple four. So the height of this triangle is going to be negative square root of three. And that means that sign of Y is going to equal to negative square root of 3/2. So just to recap we just solved for our sine X cosine X value, our cosine Y and sine of Y value. Now that we have these values, we can plug this in to the identity of cosine of X plus Y in order to solve for the value. So cosine of X plus Y is equal to cosine of X times cosine of Y minus sine of X multiplied by sign of Y. We stated that cosine of X is equal to negative 21/29. So we have negative 21/29 multiplied by cosine of Y which is one half. Then we're going to be subtracting sin of X which we stated is negative 20/29 multiplied by sine Y which is going to be negative square root 3/2. Now we just need to simplify this expression. Negative 21/29 multiplied by half is going to give us negative 21/58 and negative 20/29 multiplied by negative square root 3/2 is going to give us positive 20 square root 3/58. We can multiply these two values together to rewrite the expression as negative 21/58 minus 20 square root of 3/58. Combining these fractions together gives us the numerator of negative 21 minus 20 square root of 3/58. And we can go ahead and factor out a negative one from the numerator to leave us with negative quantity 21 plus 20 square root of 3/58. And reordering the terms in the numerator to allow the square root to be in front is going to be negative 20 square root of three plus 21/58. 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 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.

In Exercises 57–64, find the exact value of the following under the given conditions:
a. cos (α + β)
tan α = 3/4, 𝝅 < α < 3𝝅/2, and cos β = 1/4, 3𝝅/2 < β < 2𝝅

Key Concepts

Sum of Angles Formula for Cosine

Determining Trigonometric Ratios from Given Conditions

Quadrant Sign Rules for Trigonometric Functions

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