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. sin 75° + sin 15°

Accepted Answer

Hello, everyone. We are asked to rate the given trigonometric sum as a product and find the exact value. If possible, we are given the sign of 105 degrees plus the sign of 15 degrees. There are four answer choices. First, our sum to product identity for sign states, the sign of A plus, the sign of B equals two multiplied by the sign of A plus B divided by two multiplied by the cosine of A minus B divided by two. In this example, A will be 105 degrees and B is 15 degrees. So plugging those numbers into our formula, we get two multiplied by the sign of 105 plus 15 divided by two multiplied by the cosine of 105 minus 15 divided by two. Adding the numerators will have again, the sign I'm sorry, two multiplied by the sign of divided by two multiplied by the cosine of 90 divided by two. And this would be two multiplied by the sine of degrees cosine of 45 degrees. So this is part of our answer because this is rewriting it as a product. But we still want to find our exact value. So we're gonna go a couple steps further using a unit circle. I'm gonna do two multiplied by the value of the sign of degrees, which is radical three divided by two. And that's gonna be multiplied by the value of the cosine of 45 degrees, which is radical two divided by two. And when I multiply this together, I get radical six divided by two because I cross reduced one of my denominators with the leading two. And can I simplify that any farther? Nope, there is nothing I can simplify there. So the exact value is radical six divided by two. And so looking for two multiplied by the sine of 60 degrees cosine of 45 degrees and the square is six divided by two, we get answer choice D 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. sin 75° + sin 15°

Key Concepts

Sum-to-Product Formulas

Exact Values of Special Angles

Angle Addition and Subtraction

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