# What are the exact values of cos 150° and sin 150°?

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Nov 30, 2016

$\cos \left(150\right) = - \frac{\sqrt{3}}{2}$

$\sin \left(150\right) = \frac{1}{2}$

#### Explanation:

To find cos(150) use the cosine addition formula to add 60 degrees and 90 degrees.

$\cos \left(a + b\right) = \cos a \cos b \textcolor{red}{-} \sin a \sin b$

$\cos \left(90 + 60\right) = \cos 90 \cos 60 - \sin 90 \sin 60$

$\cos \left(90 + 60\right) = 0 \cdot \frac{1}{2} - 1 \cdot \frac{\sqrt{3}}{2}$

$\cos \left(150\right) = - \frac{\sqrt{3}}{2}$

For sin(150) use the sine addition formula

$\sin \left(a + b\right) = \sin a \cos b + \cos a \sin b$

$\sin \left(90 + 60\right) = \sin 90 \cos 60 + \cos 90 \sin 60$

$\sin \left(90 + 60\right) = 1 \cdot \frac{1}{2} + 0 \cdot \frac{\sqrt{3}}{2}$

$\sin \left(150\right) = + \frac{1}{2}$

Using the common angles above you can add or subtract them to determine the sine and cosine of other angles.

As an example, to determine cos(75) you could use the cosine addition formula to add 45 and 30 degrees.

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