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Question

# Find the exact value of the expression.cos (-165 degrees)

Solution

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Step 1
1 of 2

Cosine is a even function, so $\cos(-165\text{\textdegree})=\cos 165\text{\textdegree}$.

Notice that $165\text{\textdegree}=120\text{\textdegree}+45\text{\textdegree}$, so

$\cos (-165\text{\textdegree})=\cos 165\text{\textdegree}=\cos(120\text{\textdegree}+45\text{\textdegree})$

Apply formula (1):

$\cos(120\text{\textdegree}+45\text{\textdegree})=\\\\ =\cos 120\text{\textdegree}\cos 45\text{\textdegree}-\sin 120\text{\textdegree}\sin 45\text{\textdegree}=\\\\ =-\dfrac{1}{2}\cdot \dfrac{\sqrt 2}{2}-\dfrac{\sqrt 3}{2}\cdot \dfrac{\sqrt 2}{2}=\\\\ =-\dfrac{\sqrt 2}{4}-\dfrac{\sqrt 6}{4}=\\\\ =\dfrac{-\sqrt 6-\sqrt 2}{4}$

Sum and Difference Formulas for Cosine and Sine:

1. $\cos(a\pm b)=\cos a\cos b\mp \sin a\sin b$

2. $\sin (a\pm b)=\sin a\cos b\pm \cos a\sin b$

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