# How do you find the values of the other trigonometric functions of θ from the information given cos theta = − 5/13 , and theta is in quadrant III?

May 30, 2015

This comes from a $5$, $12$, $13$ triangle (${5}^{2} + {12}^{2} = {13}^{2}$)

$\sin \theta = - \frac{12}{13}$ ($\sin < 0$ in QIII)

$\tan \theta = \sin \frac{\theta}{\cos} \theta = \frac{5}{12}$

$\cot \theta = \cos \frac{\theta}{\sin} \theta = \frac{12}{5}$

$\sec \theta = \frac{1}{\cos} \theta = - \frac{13}{5}$

$\csc \theta = \frac{1}{\sin} \theta = - \frac{13}{12}$

Incidentally, the $5 - 12 - 13$ triangle is the next right angled triangle in a sequence that begins with the $3 - 4 - 5$ triangle.

${a}_{k} = 2 k + 3$

${b}_{k} = \frac{{a}_{k}^{2} - 1}{2} = 2 {k}^{2} + 6 k + 4$

${c}_{k} = \frac{{a}_{k}^{2} + 1}{2} = 2 {k}^{2} + 6 k + 5$