In Exercises 5–8, each expression is the right side of the formul...

Question

In Exercises 5–8, each expression is the right side of the formula for cos (α - β) with particular values for α and β.

b. Write the expression as the cosine of an angle.

5π π 5π π
cos ------- cos -------- + sin -------- sin -------
12 12 12 12

Accepted Answer

Hello, today we're gonna be simplifying the given trigonometric expression. So we are given cosine of nine pi over 16, multiplied by cosine of five pi over plus sine of nine pi over 16, multiplied by sine of five pi over 16. We're going to have to reduce this given trigonometric expression. And we do have one identity for cosine that helps us reduce the expression. We can use the difference property for cosine and the difference property of cosine states that if you have cosine of A minus B, this can be simplified as cosine of a multiplied by cosine of B plus sign of A multiplied by sign of B. We can reduce the fault in the given trigonometric expression by using the property going from right to left. But we're going to need to identify our A and our B values. Well, one trick in doing this is to know that the first cosine value and the first sign value have the same angle and the second cosine value and second sine value have the same angle. So if we take a look at our expression, the first cosine value given to us is nine pi over 16. And the first sign value given to us is nine pi over 16. That means we can allow A to equal to nine pi over 16. And if we take a look at the second cosine and second sign values, we have five pi over 16 for both of these values. So we can allow B to equal to five pi over 16. Now, if we use our difference property for cosine, we can reduce the given expression as cosine nine pi over 16 minus five pi over 16, nine pi over 16 minus five pi over 16 is going to reduce to give us cosine of four pi over 16 and cosine of four pi over 16 is going to reduce to give us cosine of pi over four. And this is going to be the simplified form of the given trigonometric expression. 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 5–8, each expression is the right side of the formula for cos (α - β) with particular values for α and β. b. Write the expression as the cosine of an angle. 5π π 5π π cos ------- cos -------- + sin -------- sin ------- 12 12 12 12

Key Concepts

Cosine of Angle Difference Formula

Angle Measurement in Radians

Trigonometric Identities

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