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Q. 15

Expert-verified
Found in: Page 500

### Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

# express each product as a sum containing only sines or only cosines $\mathrm{sin}\frac{3\vartheta }{2}\mathrm{cos}\frac{\vartheta }{2}$

$\mathrm{sin}\frac{3\vartheta }{2}\mathrm{cos}\frac{\vartheta }{2}=\frac{1}{2}\left[\mathrm{sin}\left(2\vartheta \right)+\mathrm{sin}\left(\vartheta \right)\right]$

See the step by step solution

## Step 1. Given expression

$\mathrm{sin}\frac{3\vartheta }{2}\mathrm{cos}\frac{\vartheta }{2}$

## Step 2. expressing them in terms of single sine or cosine

Using identity :

$\mathrm{sin}x+\mathrm{sin}y=2\mathrm{sin}\frac{x+y}{2}\mathrm{cos}\frac{x-y}{2}$

We get :

role="math" localid="1646448141774" $\mathrm{sin}\frac{3\vartheta }{2}\mathrm{cos}\frac{\vartheta }{2}=\frac{1}{2}\left[\mathrm{sin}\left(\frac{3\vartheta }{2}+\frac{\vartheta }{2}\right)+\mathrm{sin}\left(\frac{3\vartheta }{2}-\frac{\vartheta }{2}\right)\phantom{\rule{0ex}{0ex}}\mathrm{sin}\frac{3\vartheta }{2}\mathrm{cos}\frac{\vartheta }{2}=\frac{1}{2}\left[\mathrm{sin}\left(2\vartheta \right)+\mathrm{sin}\left(\vartheta \right)\right]$