# How do you find the exact value of cos 3pi/4?

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Jim G. Share
Mar 29, 2016

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

#### Explanation:

The angle $\frac{3 \pi}{4} \text{ is in the 2nd quadrant}$

where the cos ratio has a negative value. Now the related acute angle for $\frac{3 \pi}{4} \text{ is } \frac{\pi}{4}$

then $\cos \left(\frac{3 \pi}{4}\right) = - \cos \left(\frac{\pi}{4}\right)$

Using the 45-45-90 degree triangle with sides 1 , 1 , $\sqrt{2}$

where $\cos {45}^{\circ} = \cos \left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$

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

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