# How do you express sin(pi/ 4 ) * cos( ( 5 pi) / 4 )  without using products of trigonometric functions?

Apr 4, 2016
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#### Explanation:

$P = \sin \left(\frac{\pi}{4}\right) \cos \left(\frac{5 \pi}{4}\right)$
Trig table --> $\sin \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2.}$
cos ((5pi)/4 = cos (pi/4 + pi) = -cos (pi/4) = - sqrt2/2
$P = \left(- \frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{2}}{2}\right) = - \frac{1}{2}$

Apr 4, 2016

$\sin \left(\frac{\pi}{4}\right) \cos \left(\frac{5 \pi}{4}\right) = \frac{1}{2} \sin \left(\frac{3 \pi}{2}\right) - \frac{1}{2} \sin \left(\pi\right) = - \frac{1}{2}$

#### Explanation:

$2 \sin A \cos B = \sin \left(A + B\right) + \sin \left(A - B\right)$

$\sin A \cos B = \frac{1}{2} \left(\sin \left(A + B\right) + \sin \left(A - B\right)\right)$

$A = \frac{\pi}{4} , B = \frac{5 \pi}{4}$

$\sin \left(\frac{\pi}{4}\right) \cos \left(\frac{5 \pi}{4}\right) = \frac{1}{2} \left(\sin \left(\frac{\pi}{4} + \frac{5 \pi}{4}\right) + \sin \left(\frac{\pi}{4} - \frac{5 \pi}{4}\right)\right)$

$= \frac{1}{2} \left(\sin \left(\frac{6 \pi}{4}\right) + \sin \left(\frac{- 4 \pi}{4}\right)\right)$

$= \frac{1}{2} \left(\sin \left(\frac{3 \pi}{2}\right) + \sin \left(- \pi\right)\right)$

$= \frac{1}{2} \left(\sin \left(\frac{3 \pi}{2}\right) - \sin \left(\pi\right)\right)$

$= \frac{1}{2} \sin \left(\frac{3 \pi}{2}\right) - \frac{1}{2} \sin \left(\pi\right)$

$= \frac{1}{2} \cdot \left(- 1\right) - \frac{1}{2} \left(0\right)$

$= - \frac{1}{2}$