Use one or more of the six sum and difference identities to solve...

Question

Use one or more of the six sum and difference identities to solve Exercises 13–54.

In Exercises 13–24, find the exact value of each expression.

cos(135° + 30°)

Accepted Answer

Hello. Today we're gonna be determining the exact value of the given trigonometric expression. So we are given cosine of 225 degrees plus degrees. Now, in order to get the exact value, we're going to need to expand the given trigonometric expression. And we can use the sum identity of cosine to help us do that. The sum identity for cosine is given to us as cosine of A plus B is equal to cosine of a multiplied by cosine of B minus sign of A multiplied by sign of B. So what we'll need to do is we'll need to identify our A and our B value in the given trigonometric expression degrees is in the position of A and 60 degrees is in the position of B. So we're going to let a equal to 225 degrees. And we're going to let B equal to 60 degrees. Now that we have our AM B value. If we use the identity to expand the trigonometric expression, we can rewrite the expression as cosine of A which is degrees multiplied by cosine of B, which is 60 degrees minus sine of A, which is 225 degrees multiplied by sine of B which is degrees. Now, what we wanna do is we want to evaluate each quantity of cosine and sine. And in order to do that, we're going to need to use our unit circle. So let's first identify where 225 degrees is located on the unit circle. 225 degrees is located in the third quadrant. And the terminal point for 225 degrees is given to us as negative square root 2/2 comma negative square root 2/2. Next, let's identify where the angle of 60 degrees is located at 60 degrees is located in the first quadrant and the terminal point for 60 degrees is going to be positive one half comma positive square root 3/2. Now recall that any terminal point on the unit circle is defined by cosine comma sine, cosine represents any X value for any terminal point and sine represents any Y value for any terminal point. So if we are looking for cosine of 225 degrees, we want to look at the X value for the terminal point of 225 degrees. The X value for that point is going to be negative square root 2/2. So we can replace cosine 225 degrees with negative square root 2/2. Then we're going to multiply that by the X value of cosine degrees. So we look at the angle 60 degrees and the X value for that terminal point is positive one half. So cosine of 60 degrees is positive one half, then we're going to be subtracting the product of sine of 225 degrees. Now sine is every Y value for the terminal point and degrees has a Y value of negative square root 2/2. So S sign of 225 degrees is negative square root 2/2. And we're multiplying that value by sine of 60 which is going to have the value of square root 3/2. Now, what we wanna do is we want to multiply these values together negative square root 2/2, multiplied by a half is going to give us negative square root 2/4 and negative square root 2/2, multiplied by square root 3/2 is going to simplify to give us negative square root 6/4. We have a negative one times negative square root 6/4. So we can rewrite the statement as negative square root 2/ plus square root 6/4. And if we add these values together, we can rewrite this expression as negative square root two plus square root of 6/4. Let's go ahead and reorder the terms. So that way the positive term will be the leading term in the numerator. And the simplified form of the answer is going to be the square root of six minus the square root of two, all over four. This is going to be the exact value to the given trigonometric expression. 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.

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Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. cos(135° + 30°)