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

Question

Be sure that you've familiarized yourself with the first set of formulas presented in this section by working C1–C4 in the Concept and Vocabulary Check. In Exercises 1–8, use the appropriate formula to express each product as a sum or difference.
$\cos \frac{3 x}{2} \sin \frac{x}{2}$

$\cos \frac{3 x}{2} \sin \frac{x}{2}$

Accepted Answer

Hello, everyone. We are asked to convert the given trigonometric expression into an equivalent expression using a product to some identity. We are given the cosine of 103 X divided by two, multiplied by the sign of 101 X divided by two, we have four answer choices. So looking at my identities, I am looking for one that is the cosine of an angle multiplied by the sign of another angle. And I recall that we have one that is the cosine of a multiplied by the sign of B and that would equal one half multiplied by. And I'm putting it in square brackets, the sign of A plus B closed parentheses minus the sign of A minus B in parentheses, closing my square bracket. So using this formula, we're gonna say for our problem, A is 103 X divided by two and B is 101 X divided by two. So I'm gonna plug those in for A and B in my formula. So this will equal one half multiplied by a square bracket. And then the sign of A which is X divided by two plus B So X divided by two closed parentheses minus the sign of A, which again was 103 X divided by two minus B. So all 101 X divided by two closed parentheses, close square brackets, I'm going to add and subtract the numbers that are inside my parentheses. So my A's and B's and we have this equals one half multiplied by the square bracket. And inside the sign of when I add this together, I get 204 X divided by two closed parentheses, minus the sign of, let's see, 103 X minus 101 X is two X divided by two closed parentheses, closed square bracket. And now I want to simplify the fractions, reduce them if I can and it looks like I can. So we have one half multiplied by a square bracket. And inside we have the sign of, let's see, this looks like it'd be 102 X closed parentheses minus the sign of one X or just X closed parentheses, close square brackets. And that looks like it. That is as far as this goes, this is all we can simplify. So one half multiplied by the square brackets, sign 101 X minus the sign of X. And that is answer choice. D have a nice day.

Be sure that you've familiarized yourself with the first set of formulas presented in this section by working C1–C4 in the Concept and Vocabulary Check. In Exercises 1–8, use the appropriate formula to express each product as a sum or difference.
$\cos \frac{3 x}{2} \sin \frac{x}{2}$

Key Concepts

Product-to-Sum Formulas

Trigonometric Function Arguments and Angles

Simplification of Trigonometric Expressions

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