If $\theta=-\frac{5 \pi}{3}$, then find exact values for the following. If the trigonometric function is undefined for $\theta=-\frac{5 \pi}{3}$, enter DNE. $\sec (\theta)$ equals $\csc (\theta)$ equals $\tan (\theta)$ equals $\cot (\theta)$ equals No decimals

Step 1 ：First, we need to convert the angle $\theta=-\frac{5 \pi}{3}$ to an angle between $0$ and $2\pi$. We can do this by adding $2\pi$ to $\theta$ until it falls within this range. So, $\theta=-\frac{5 \pi}{3}+2\pi=-\frac{5 \pi}{3}+\frac{6 \pi}{3}=\frac{\pi}{3}$.

Step 2 ：Now, we can find the exact values of the trigonometric functions for $\theta=\frac{\pi}{3}$.

Step 3 ：The secant function, $\sec(\theta)$, is the reciprocal of the cosine function. So, $\sec(\theta)=\frac{1}{\cos(\theta)}$. We know that $\cos(\frac{\pi}{3})=\frac{1}{2}$, so $\sec(\frac{\pi}{3})=\frac{1}{\frac{1}{2}}=2$.

Step 4 ：The cosecant function, $\csc(\theta)$, is the reciprocal of the sine function. So, $\csc(\theta)=\frac{1}{\sin(\theta)}$. We know that $\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$, so $\csc(\frac{\pi}{3})=\frac{1}{\frac{\sqrt{3}}{2}}=\frac{2}{\sqrt{3}}$. To rationalize the denominator, we multiply the numerator and denominator by $\sqrt{3}$ to get $\csc(\frac{\pi}{3})=\frac{2\sqrt{3}}{3}$.

Step 5 ：The tangent function, $\tan(\theta)$, is the ratio of the sine function to the cosine function. So, $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$. We know that $\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$ and $\cos(\frac{\pi}{3})=\frac{1}{2}$, so $\tan(\frac{\pi}{3})=\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}=\sqrt{3}$.

Step 6 ：The cotangent function, $\cot(\theta)$, is the reciprocal of the tangent function. So, $\cot(\theta)=\frac{1}{\tan(\theta)}$. We know that $\tan(\frac{\pi}{3})=\sqrt{3}$, so $\cot(\frac{\pi}{3})=\frac{1}{\sqrt{3}}$. To rationalize the denominator, we multiply the numerator and denominator by $\sqrt{3}$ to get $\cot(\frac{\pi}{3})=\frac{\sqrt{3}}{3}$.

Step 7 ：So, the exact values for the trigonometric functions are $\sec(\theta)=\boxed{2}$, $\csc(\theta)=\boxed{\frac{2\sqrt{3}}{3}}$, $\tan(\theta)=\boxed{\sqrt{3}}$, and $\cot(\theta)=\boxed{\frac{\sqrt{3}}{3}}$.