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Q. 5

Expert-verified
Found in: Page 500

### Precalculus Enhanced with Graphing Utilities

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

# find the exact value of the expression cos 255$°$ - cos 195$°$

cos255$°$-cos195$°==0.7071067812$

See the step by step solution

## Step 1. Rewriting the expression

$\mathrm{cos}\left(270°-x\right)=-\mathrm{sin}x\phantom{\rule{0ex}{0ex}}\mathrm{cos}\left(180°+x\right)=-\mathrm{cos}x\phantom{\rule{0ex}{0ex}}\mathrm{cos}255°=\mathrm{cos}\left(270°-15°\right)=-\mathrm{sin}15°\phantom{\rule{0ex}{0ex}}\mathrm{cos}195°=\mathrm{cos}\left(180°+15°\right)=-\mathrm{cos}15°$

## Step 2. Calculations

Now our expression becomes

$=\mathrm{cos}255°-\mathrm{cos}195=\mathrm{cos}15°-\mathrm{sin}15°\phantom{\rule{0ex}{0ex}}Nowweknowthat\phantom{\rule{0ex}{0ex}}\mathrm{cos}15°=\mathrm{sin}75°\phantom{\rule{0ex}{0ex}}So,\phantom{\rule{0ex}{0ex}}\mathrm{cos}15°-\mathrm{sin}15°=\mathrm{sin}75°-\mathrm{sin}15°\phantom{\rule{0ex}{0ex}}\mathrm{sin}x-\mathrm{sin}y=2\mathrm{sin}\left(\frac{x-y}{2}\right)\mathrm{cos}\left(\frac{x+y}{2}\right)\phantom{\rule{0ex}{0ex}}So,\mathrm{sin}75°-\mathrm{sin}15°=2\left(\mathrm{sin}30°\mathrm{cos}45°\right)=2\frac{1}{2}\frac{1}{\sqrt{2}}\phantom{\rule{0ex}{0ex}}=0.7071067812\phantom{\rule{0ex}{0ex}}$