How do you use csctheta=5 to find cottheta?

Mar 27, 2017

$\cot \theta = \pm 2 \sqrt{6}$

Explanation:

The terminal ray of an angle $\theta$ in standard position intersects a point $\left(x , y\right)$ on the unit circle such that $\csc \theta = \frac{1}{y}$, $\cot \theta = \frac{x}{y}$, and ${x}^{2} + {y}^{2} = 1$.

Knowing that $\csc \theta = 5$ we know that $y = \frac{1}{5}$

Since ${x}^{2} + {y}^{2} = 1$, $x = \pm \sqrt{1 - {y}^{2}}$

$\cot \theta = \frac{x}{y} = \frac{\pm \sqrt{1 - {y}^{2}}}{y} = \frac{\pm \sqrt{1 - {\left(\frac{1}{5}\right)}^{2}}}{\frac{1}{5}}$
$\cot \theta = \pm 5 \sqrt{1 - \frac{1}{25}} = \pm 5 \sqrt{\frac{24}{25}} = \pm \cancel{5} \frac{\sqrt{24}}{\cancel{5}} = \pm 2 \sqrt{6}$

Mar 27, 2017

$\pm 2 \sqrt{6}$

Explanation:

Use trig identity:
$1 + {\cot}^{2} t = {\csc}^{2} t$
In this case:
$1 + {\cot}^{2} t = 25$
${\cot}^{2} t = 24$
$\cot t = \pm 2 \sqrt{6}$