Cosine of 120 Degrees

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Theorem

$\cos 120^\circ = \cos \dfrac {2 \pi} 3 = - \dfrac 1 2$

where $\cos$ denotes cosine.


Proof

\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \cos 120^\circ\) \(=\) \(\displaystyle \) \(\) \(\displaystyle \) \(\displaystyle \cos \left({90^\circ + 30^\circ}\right)\) \(\displaystyle \) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\) \(\displaystyle \) \(\displaystyle - \sin 30^\circ\) \(\displaystyle \) \(\displaystyle \)          Cosine of Angle plus Right Angle          
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\) \(\displaystyle \) \(\displaystyle - \frac 1 2\) \(\displaystyle \) \(\displaystyle \)          Sine of 30 Degrees          

$\blacksquare$


Sources