# If cot theta = - 2 and cos theta < 0, how do you find sintheta?

Mar 30, 2018

$1 + {\cot}^{2} \left(\theta\right) = {\csc}^{2} \left(\theta\right)$

#### Explanation:

Substitute ${\cot}^{2} \left(\theta\right) = {\left(- 2\right)}^{2}$:

$1 + {\left(- 2\right)}^{2} = {\csc}^{2} \left(\theta\right)$

$5 = {\csc}^{2} \left(\theta\right)$

Substitute ${\csc}^{2} \left(\theta\right) = \frac{1}{\sin} ^ 2 \left(\theta\right)$

$5 = \frac{1}{\sin} ^ 2 \left(\theta\right)$

${\sin}^{2} \left(\theta\right) = \frac{1}{5}$

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

Because we are told that $\cos \left(\theta\right) < 0$ and $\cot \left(\theta\right) = - 2$, we know that the sine function must be positive in this quadrant:

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