Question

if $\theta = \frac{-1\pi}{3}$, then $\sin(\theta) = $ $\cos(\theta) = $ give exact values. no decimals allowed!

Explanation:

Step1: Use the property of sine function

The sine function is odd, i.e., \(\sin(-x)=-\sin(x)\).
For \(\theta =-\frac{\pi}{3}\), \(\sin(-\frac{\pi}{3})=-\sin(\frac{\pi}{3})\).
Since \(\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}\), then \(\sin(-\frac{\pi}{3})=-\frac{\sqrt{3}}{2}\).

Step2: Use the property of cosine function

The cosine function is even, i.e., \(\cos(-x)=\cos(x)\).
For \(\theta =-\frac{\pi}{3}\), \(\cos(-\frac{\pi}{3})=\cos(\frac{\pi}{3})\).
Since \(\cos(\frac{\pi}{3})=\frac{1}{2}\), then \(\cos(-\frac{\pi}{3})=\frac{1}{2}\).

Answer:

\(\sin(\theta)=-\frac{\sqrt{3}}{2}\), \(\cos(\theta)=\frac{1}{2}\)