# How do you determine sintheta given sectheta=-7/5, 180^circ<theta<270^circ?

Nov 16, 2016

#### Explanation:

The secant is the reciprocal of the cosine:

$\cos \left(\theta\right) = \frac{1}{\sec} \left(\theta\right) = - \frac{5}{7}$

This is a variant of ${\sin}^{2} \left(\theta\right) + {\cos}^{2} \left(\theta\right) = 1$

$\sin \left(\theta\right) = \pm \sqrt{1 - {\cos}^{2} \left(\theta\right)}$

Substitute $\frac{25}{49}$ for ${\cos}^{2} \left(\theta\right)$

$\sin \left(\theta\right) = \pm \sqrt{1 - \frac{25}{49}}$

$\sin \left(\theta\right) = \pm \sqrt{\frac{24}{49}}$

$\sin \left(\theta\right) = \pm \frac{\sqrt{24}}{7}$

We are told that we are in the third quadrant so use the negative value:

$\sin \left(\theta\right) = - \frac{\sqrt{24}}{7}$