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.

c. Find the exact value of the expression.

cos 50° cos 20° + sin 50° sin 20°

Accepted Answer

Hello, today we're gonna be finding the exact value of the trigonometric expression. So what we are given is cosine of 70 degrees multiplied by cosine of 10 degrees plus sine of 70 degrees multiplied by sine of 10 degrees. So we're going to need to first simplify the given expression. And we do have a method to help us simplify that we can use the difference property of cosine the difference property of cosine states that if you have cosine A minus B, this can be rewritten as cosine of a multiplied by cosine of B plus sine of A multiplied by sign of B. So we're going to be using this property going from right to left. But in order to simplify the expression, let's first identify our A and our B values. Now, a trick into doing this is to notice that in the property, the first cosine and sine values have the same angle and the second cosine and sine value have the same angle. If we take a look at the expression, the first cosine and sine values have an angle of 70 degrees. So we're going to allow A to equal to 70 degrees and the second cosine and sine values have an angle of 10 degrees. So we're going to allow B to equal to 10 degrees. Now, if we use our property, we can simplify the expression to cosine of 70 minus 10 and 70 minus 10 is going to give us 60. So we have cosine of 60 degrees. Now, what we want to do in order to evaluate cosine to 60 degrees is we want to find 60 degrees on the unit circle, 60 degrees on the unit circle is going to exist in quadrant one and the terminal point for 60 degrees is going to be one half, comma square root 3/2. And recall that any terminal point on the unit circle is defined as cosine commas sign. What this means is that cosine represents any X value of any terminal point on the unit circle. And sine represents any Y value of any terminal point on the unit circle. So since we are looking for cosine of 60 degrees, we want to look at the X value for the terminal point at 60 degrees and that X value is going to be one half. So that means cosine of 60 degrees is equal to a half. And this is going to be the exact value for the given trigonometric expression. 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 5–8, each expression is the right side of the formula for cos (α - β) with particular values for α and β. c. Find the exact value of the expression. cos 50° cos 20° + sin 50° sin 20°

Key Concepts

Cosine of Angle Difference Formula

Trigonometric Values

Exact Value Calculation

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