Answer
$sin\theta=-\frac{12}{13}$,
$cos\theta=-\frac{5}{13}$,
$tan\theta=\frac{12}{5}$,
$cot\theta=\frac{5}{12}$,
$csc\theta=-\frac{13}{12}$,
$sec\theta=-\frac{13}{5}$
Work Step by Step
Step 1. Given $sec\theta=\frac{1}{cos\theta}=-\frac{13}{5}$ and $tan\theta\gt0$, we can identify that $\theta$ is in Quadrant III.
Step 2. From the above given condition, we get $cos\theta=-\frac{5}{13}$,
Step 3. Similarly, $sin\theta=-\sqrt {1-cos^2\theta}=-\frac{12}{13}$ and $csc\theta=\frac{1}{sin\theta}=-\frac{13}{12}$
Step 4. Use the results above $tan\theta=\frac{sin\theta}{cos\theta}=\frac{12}{5}$ and $cot\theta=\frac{1}{tan\theta}=\frac{5}{12}$
