Question

what is cos \frac{4\pi}{3}?

Explanation:

Step1: Convert angle to standard position

The angle \(\frac{4\pi}{3}\) can be written as \(\pi+\frac{\pi}{3}\).

Step2: Use cosine addition formula

\(\cos(A + B)=\cos A\cos B-\sin A\sin B\). Here \(A=\pi\), \(B = \frac{\pi}{3}\).
\(\cos(\pi+\frac{\pi}{3})=\cos\pi\cos\frac{\pi}{3}-\sin\pi\sin\frac{\pi}{3}\)
Since \(\cos\pi=- 1\), \(\sin\pi = 0\), \(\cos\frac{\pi}{3}=\frac{1}{2}\), \(\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}\)

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Answer:

\(-\frac{1}{2}\)