Answer
$\sin\theta=\dfrac{5}{13}$
$\cos\theta=\dfrac{12}{13}$
$\tan\theta=\dfrac{5}{12}$
$\csc\theta=\dfrac{13}{5}$
$\sec\theta=\dfrac{13}{12}$
$\cot\theta=\dfrac{12}{5}$
Work Step by Step
First, we have to calculate the third side of the triangle. We can use the Pythagorean Theorem:
$$c^2=a^2+b^2 \quad\text{ where $c$ is the hypotenuse}$$.
$5^2+12^2=c^2$
$25+144=c^2$
$169=c^2$
$13=c$
Now we can calculate all six trigonometric functions for $\theta$ using their definitions:
$\sin\theta=\dfrac{\text{Opposite}}{\text{Hypotenuse}}=\dfrac{5}{13}$
$\cos\theta=\dfrac{\text{Adjacent}}{\text{Hypotenuse}}=\dfrac{12}{13}$
$\tan\theta=\dfrac{\text{Opposite}}{\text{Adjacent}}=\dfrac{5}{12}$
$\csc\theta=\dfrac{\text{Hypotenuse}}{\text{Opposite}}=\dfrac{13}{5}$
$\sec\theta=\dfrac{\text{Hypotenuse}}{\text{Adjacent}}=\dfrac{13}{12}$
$\cot\theta=\dfrac{\text{Adjacent}}{\text{Opposite}}=\dfrac{12}{5}$
