evaluate: $secleft(\frac{5pi}{4}\right)$\n$sqrt{3}$\n$\frac{2sqrt{3}}{3}$\n$-sqrt{2}$\n$\frac{sqrt{3}}{3}$\nd…

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: Find the reference - angle

The angle $\frac{5\pi}{4}$ is in the third - quadrant. The reference angle $\theta_{r}=\frac{5\pi}{4}-\pi=\frac{\pi}{4}$. 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:

$-\sqrt{2}$