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Q. 1

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:$\mathrm{sin}195°·\mathrm{cos}75°$

The exact value is $\frac{1}{2}\left[\frac{\sqrt{3}}{2}-1\right]$.

See the step by step solution

Step 1. Given information.

Consider the given question,

$\mathrm{sin}195°·\mathrm{cos}75°$

We know localid="1646458669499" role="math" $\mathrm{sin}a·\mathrm{sin}b=\frac{1}{2}\left[\mathrm{sin}\left(a+b\right)+\mathrm{sin}\left(a-b\right)\right]$.

Using the formula, we get,

$\mathrm{sin}195°·\mathrm{cos}75°=\frac{1}{2}\left[\mathrm{sin}\left(195°+75°\right)+\mathrm{sin}\left(195°-75°\right)\right]$

Step 2. Simplify the equation.

Continuing the above equation,

$\mathrm{sin}195°·\mathrm{cos}75°=\frac{1}{2}\left[\mathrm{sin}\left(270°\right)+\mathrm{sin}\left(120°\right)\right]\phantom{\rule{0ex}{0ex}}\mathrm{sin}195°·\mathrm{cos}75°=\frac{1}{2}\left[-1+\frac{\sqrt{3}}{2}\right]\phantom{\rule{0ex}{0ex}}\mathrm{sin}195°·\mathrm{cos}75°=\frac{1}{2}\left[\frac{\sqrt{3}}{2}-1\right]$