# Question 93462

Jan 30, 2017

$\tan \left(\theta\right) = \frac{5}{3}$
$\sec \left(\theta\right) = - \frac{\sqrt{34}}{3}$
$\cos \left(\theta\right) = - \frac{3 \sqrt{34}}{34}$
$\sin \left(\theta\right) = - \frac{5 \sqrt{34}}{34}$
$\csc \left(\theta\right) = - \frac{\sqrt{34}}{5}$

#### Explanation:

Given: $\cot \left(\theta\right) = \frac{3}{5} \mathmr{and} \sec \left(\theta\right) < 0$

Use the identity $\tan \left(\theta\right) = \frac{1}{\cot} \left(\theta\right)$:

$\tan \left(\theta\right) = \frac{1}{\frac{3}{5}}$

$\tan \left(\theta\right) = \frac{5}{3}$

Use the identity $1 + {\tan}^{2} \left(\theta\right) = {\sec}^{2} \left(\theta\right)$

$1 + {\left(\frac{5}{3}\right)}^{2} = {\sec}^{2} \left(\theta\right)$

${\sec}^{2} \left(\theta\right) = \frac{34}{9}$

$\sec \left(\theta\right) = - \frac{\sqrt{34}}{3}$

Use the identity $\cos \left(\theta\right) = \frac{1}{\sec} \left(\theta\right)$

$\cos \left(\theta\right) = - \frac{3}{\sqrt{34}}$

$\cos \left(\theta\right) = - \frac{3 \sqrt{34}}{34}$

Use the identity $\tan \left(\theta\right) = \sin \frac{\theta}{\cos} \left(\theta\right)$

$\tan \left(\theta\right) \cos \left(\theta\right) = \sin \left(\theta\right)$

$\sin \left(\theta\right) = \frac{5}{3} \left(- \frac{3 \sqrt{34}}{34}\right)$

$\sin \left(\theta\right) = \frac{- 5 \sqrt{34}}{34}$

Use the identity $\csc \left(\theta\right) = \frac{1}{\sin} \left(\theta\right)$

csc(theta) = 1/(-5/sqrt(34)#

$\csc \left(\theta\right) = - \frac{\sqrt{34}}{5}$