Question

6 multiple choice 1 point evaluate without using a calculator by using ratios in a reference triangle. cos(-\frac{5\pi}{4}) \frac{\sqrt{3}}{2} -\frac{\sqrt{2}}{2} -\frac{\sqrt{3}}{2} \frac{1}{2}

Explanation:

Step1: Use the property of cosine function

Since \(\cos(-x)=\cos(x)\), so \(\cos(-\frac{5\pi}{4})=\cos(\frac{5\pi}{4})\).

Step2: Find the reference angle

\(\frac{5\pi}{4}=\pi+\frac{\pi}{4}\). The reference angle \(\theta'=\frac{5\pi}{4}-\pi=\frac{\pi}{4}\).

Step3: Determine the sign

\(\frac{5\pi}{4}\) is in the third - quadrant. In the third - quadrant, \(\cos(x)<0\).

Step4: Use the cosine ratio of the reference angle

We know that \(\cos(\frac{\pi}{4})=\frac{\sqrt{2}}{2}\). So \(\cos(\frac{5\pi}{4})=-\cos(\frac{\pi}{4})\) (because of the sign in the third - quadrant).

Answer:

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