evaluate: sec(5π/4) √3 2√3/3 -√2 √3/3 done

Explanation:

Step1: Recall secant - cosine relationship

$\sec(x)=\frac{1}{\cos(x)}$, so $\sec(\frac{5\pi}{4})=\frac{1}{\cos(\frac{5\pi}{4})}$.

Step2: Determine the reference angle

The angle $\frac{5\pi}{4}$ is in the third - quadrant. The reference angle $\theta'=\frac{5\pi}{4}-\pi=\frac{\pi}{4}$. In the third - quadrant, cosine is negative. And $\cos(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$, so $\cos(\frac{5\pi}{4})=-\frac{\sqrt{2}}{2}$.

Step3: Calculate the secant value

Substitute $\cos(\frac{5\pi}{4}) = -\frac{\sqrt{2}}{2}$ into the secant formula: $\sec(\frac{5\pi}{4})=\frac{1}{\cos(\frac{5\pi}{4})}=\frac{1}{-\frac{\sqrt{2}}{2}}=-\sqrt{2}$.

Answer:

$-\sqrt{2}$