# How do you find the remaining trigonometric functions of theta given cottheta-=1/2 and costheta>0?

Jul 7, 2017

$\cot t = \frac{1}{2}$ --> tan t = 2
Use trig identity:
${\cos}^{2} t = \frac{1}{1 + {\tan}^{2} t} = \frac{1}{1 + 4} = \frac{1}{5}$
$\cos t = \frac{1}{\sqrt{5}} = \frac{\sqrt{5}}{5}$ --> (because cos t > 0)
${\sin}^{2} t = 1 - {\cos}^{2} t = 1 - \frac{1}{5} = \frac{4}{5}$
$\sin t = \pm \frac{2}{\sqrt{5}} = \pm \frac{2 \sqrt{5}}{5}$
tan t = 2 , then, t is in Quadrant 1, and sin t is positive
$\sin t = \frac{2 \sqrt{5}}{5}$
$\sec t = \frac{1}{\cos t} = \sqrt{5}$
$\csc t = \frac{1}{\sin t} = \frac{\sqrt{5}}{2}$