In Exercises 35–38, find the exact value of the following under t...

Question

In Exercises 35–38, find the exact value of the following under the given conditions:

b. cos(α﹣β)

3 𝝅 12 𝝅 
sin α = ------- , 0 < α < -------- , and sin β = --------- , --------- < β < 𝝅. 
5 2 13 2

Accepted Answer

Hello, today we're gonna be solving for the exact value of the given expression. So what we are given is cosine of A minus B. Now, before we solve the exact value, we can first expand cosine of A minus B, we can expand this expression by using the difference identity for cosine and the difference identity for cosine states that we can expand cosine of A minus B as cosine of A multiplied by cosine of B plus sign of A multiplied by sign of B. So what we'll need to do is we'll need to solve our cosine A cosine B sign of A and S sign of B values. In order to find the exact value of cosine A minus B, we can solve for these values by using the given conditions. So let's take a look at the first condition. The first condition states that sign of A is equal to 7/25. And the angle of A exists within the region of the unit circle between zero and pi over two. One thing to note is that the region of the unit circle between zero and pi over two is going to be quadrant one and within quadrant one, any X values or Y values are going to be positive. So since we are given sign of a, how can we use this value to solve for cosine of a? Well, we can use a right triangle that is located in quadrant one. Now recall that the identity for sine of any right triangle is defined as the height of the triangle over its hypotenuse. And the identity for cosine of any right triangle is defined as the length of the triangle over its hypotenuse. So using the identity of sine and the given value for sign of a, we can label the triangle with the height of seven and a hypotenuse of 25. So what we need to do now is we need to solve for the length of the triangle. In order to get our cosine value, we can solve for this length by using the Pythagorean theorem. And the Pythagorean theorem states that the hypo squared is equal to the length squared plus the height squared. In this case, H will be 25 Y will be seven and plugging this into the equation will give us 25 squared is equal to X squared plus seven squared. 25 squared will simplify to give us and seven squared will give us the value of 49 will subtract 49 to both sides of the equation. Leaving us with X squared is equal to 576. And taking the square root of both sides of the equation will give us X is equal to 24. So that means that the right triangle is going to have a length of positive 24. Now that we have the length, we can use the identity for cosine to get the value of cosine of A. And in this case here, cosine of A is going to equal to 24/25. So we have our sign of a value and a cosine of a value we need to solve for sign of B and cosine of B value. Let's take a look at the second condition given to us, we are given that sign of B is equal to 9/41. And we are told that the angle B exists within the region of the unit circle between Pi over two and pi. Now, one thing to note is that the region of the unit circle between pi over two and pi is going to be quadrant two. And within quadrant two, any X value is negative and any Y value is positive. So in order to get our cosine value, we're going to use the same method as before and use a right triangle. But this time located in quadrant two using the identity of sine and the given sign value, we can assign the height of the triangle as nine and the hypotenuse as 41 and then we have to solve for the length of the triangle. So in this case, our H is going to be 41 our Y is going to be nine. And if we plug this into the Pythagorean theorem, we get 41 squared is equal to X squared plus nine squared. 41 squared will give us the value of and nine squared will give us the value of 81. In order to solve for X, we subtract 81 to both sides of the equation. Leaving us with X squared is equal to 1600. And taking the square root of both sides will give us the value of 40. So the length of this triangle is going to be 40. But keep in mind that any X value within this region of quadrant two is going to be negative. So the length of the triangle is going to be negative 40. And using the identity for cosine of a right triangle, cosine of B will equal to negative 40/41. So now that we have the values for cosine of a cosine of B sign of A and sign of B, we can plug this in to the difference identity for cosine. So we can find the exact value of cosine A minus B. So if we plug this into the identity, we can rewrite it as cosigner A which is negative 20 I'm sorry, positive 24/25 multiplied by cosine of B which is negative 40/41 plus sign of A which was given to us as 7/25 multiplied by sign of B which was given to us as 9/41. Now, what we need to do is multiply the values together 24/25 multiplied by negative 40. Over 41 is going to give us the value of negative 960 over and 7/25 multiplied by nine. Over 41 will give us the value of 63 over 1025 and adding these two fractions together is going to give us the value of negative 897 over 1025. So what this means is that the exact value of cosine A minus B is going to be negative over 1025. And this is going to be the answer to this problem. And with that being said, the solution 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 35–38, find the exact value of the following under the given conditions: b. cos(α﹣β) 3 𝝅 12 𝝅 sin α = ------- , 0 < α < -------- , and sin β = --------- , --------- < β < 𝝅. 5 2 13 2

Key Concepts

Trigonometric Identities

Sine and Cosine Values

Angle Restrictions

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