Be sure that you've familiarized yourself with the second set ...

Question

Be sure that you've familiarized yourself with the second set of formulas presented in this section by working C5–C8 in the Concept and Vocabulary Check. In Exercises 9–22, express each sum or difference as a product. If possible, find this product's exact value. cos 75° ﹣ cos 15°

Accepted Answer

Hello, everyone. We are asked to rate the given trigonometric difference as a product and find the exact value. If possible, we are given the cosine of 105 degrees minus the cosine of 15 degrees. Our answer choices are a negative two sine, 45 degrees cosine of 60 degrees. And the value is negative radical two divided by two B, two sine of 120 degrees cosine of 90 degrees with an exact value of zero C negative two sine of 60 degrees sine of 45 degrees. And the exact value is negative radical six divided by two D two sine of 60 degrees sine of 45 degrees and radical six divided by two is the exact value. First, I recall that if I want to write the difference of cosines as a product, the formula for that is that the cosine of A minus, the cosine of B will equal negative two sign of A plus B divided by two in parentheses, multiplied by the sign of A minus B divided by two in parentheses. So in our example, A is 105 degrees B is 15 degrees. So I'm going to plug those values into our formula. So I have negative two multiplied by the sine of 105 degrees plus 15 degrees in parentheses, divided by two multiplied by the sign of as a fraction in parentheses, 105 degrees minus 15 degrees. Again, divided by two. Simplifying my fractions, I have negative two multiplied by the sign of in parentheses. 120 degrees divided by two closed parentheses multiplied by the sine of 90 degrees, divided by two closed parentheses. So we have negative two multiplied by the sine of 60 degrees sine of 45 degrees. So that is our difference written as a product. So this is part of our answer. Negative two multiplied by the sine of 60 degrees. S sign of 45 degrees. Then to find the exact value, we're gonna use a unit circle and find the value that matches the sign of 60 degrees. And when we do so, we have negative two multiplied by radical three divided by two multiplied by the exact value of the sine of 45 degrees from our unit circle which is radical two divided by two. And when I cross reduce, I can cancel out one set of twos and I'm left with negative radical six divided by two. And when I check my answer choices, this matches answer choice C so negative two sine of degrees sin of 45 degrees. And the exact value is negative radical six divided by two have a nice day.

Be sure that you've familiarized yourself with the second set of formulas presented in this section by working C5–C8 in the Concept and Vocabulary Check. In Exercises 9–22, express each sum or difference as a product. If possible, find this product's exact value. cos 75° ﹣ cos 15°

Key Concepts

Sum-to-Product Formulas

Exact Values of Special Angles

Angle Addition and Subtraction

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