# Suppose α is an angle in standard position whose terminal side lies in Quadrant III. If tan α = 2 3 , find the value of sin α?

May 11, 2018

$\sin \left(\alpha\right) = - 0.999$

#### Explanation:

$\tan \alpha = 23$ ; $\alpha = {87.5}^{o} + n \left(180\right)$

Standard position means one side is along the x-axis, with vertex at the origin.
Terminal side in Quadrant III is $180 \le \alpha \le 270$

$\alpha = {87.5}^{o} + \left(180\right) = {267.5}^{0}$

$\sin \left(267.5\right) = - 0.999$

https://mathbitsnotebook.com/Algebra2/TrigConcepts/TCStandardPosition.html

May 11, 2018

$\sin a = - \frac{23}{\sqrt{530}}$

#### Explanation:

tan a = 23 --> $\cot a = \frac{1}{23}$
${\sin}^{2} a = \frac{1}{1 + {\cot}^{2} a} = \frac{1}{1 + \frac{1}{529}} = \frac{529}{530}$
$\sin a = \pm \frac{23}{\sqrt{530}}$
Since a in in Quadrant 3, sin a is negative
$\sin a = - \frac{23}{23.05} = - 0.999$