Question

evaluate the expression \\( \sin ^ { - 1 } \left( \cos \left( \frac { 5 \pi } { 3 } \
ight) \
ight) \\).

give your answer as an exact value

Explanation:

Step1: Find the value of \(\cos(\frac{5\pi}{3})\)

Using the unit - circle, \(\cos(\frac{5\pi}{3})=\cos(2\pi-\frac{\pi}{3})\).
Since \(\cos(A - B)=\cos A\cos B+\sin A\sin B\) and \(\cos(2\pi)=1,\sin(2\pi) = 0\), we have \(\cos(2\pi-\frac{\pi}{3})=\cos\frac{\pi}{3}\).
So, \(\cos(\frac{5\pi}{3})=\frac{1}{2}\).

Step2: Evaluate \(\sin^{-1}(\frac{1}{2})\)

Let \(y = \sin^{-1}(x)\), then \(y\in[-\frac{\pi}{2},\frac{\pi}{2}]\) and \(\sin y=x\).
We want to find \(y\) such that \(\sin y=\frac{1}{2}\) and \(y\in[-\frac{\pi}{2},\frac{\pi}{2}]\).
The solution of the equation \(\sin y=\frac{1}{2}\) in the interval \([-\frac{\pi}{2},\frac{\pi}{2}]\) is \(y=\frac{\pi}{6}\).

Answer:

\(\frac{\pi}{6}\)