Question

if ( \theta=\frac{-5 pi}{4} ), then ( sin (\theta)= ) ( cos (\theta)= ) give exact values. no decimals allowed!

Explanation:

Step1: Find the coterminal positive angle

Since \(\theta =-\frac{5\pi}{4}\), we add \(2\pi\) (because \(2\pi\) is the period of sine and cosine functions) to get a coterminal positive angle.
\(\theta_{positive}=-\frac{5\pi}{4}+2\pi=\frac{-5\pi + 8\pi}{4}=\frac{3\pi}{4}\)

Step2: Determine the reference angle

For \(\theta=\frac{3\pi}{4}\), which is in the second quadrant. The reference angle \(\theta_{r}=\pi-\frac{3\pi}{4}=\frac{\pi}{4}\)

Step3: Find \(\sin(\theta)\)

In the second quadrant, \(\sin\theta> 0\). Using the unit - circle values, \(\sin(\frac{\pi}{4})=\frac{\sqrt{2}}{2}\), so \(\sin(\theta)=\sin(\frac{3\pi}{4})=\frac{\sqrt{2}}{2}\)

Step4: Find \(\cos(\theta)\)

In the second quadrant, \(\cos\theta<0\). Using the unit - circle values, \(\cos(\frac{\pi}{4})=\frac{\sqrt{2}}{2}\), so \(\cos(\theta)=\cos(\frac{3\pi}{4})=-\frac{\sqrt{2}}{2}\)

Answer:

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