Question #39687

Jul 2, 2017

$\sin \left(\theta\right) = \frac{4}{5}$

Explanation:

Obtuse means that the angle is between 90˚ and 180˚.

$90 < \left(\theta\right) < 180$

We know the following is true for any angle $\theta$, so substitute in the known information and solve for $\sin \left(\theta\right)$:

${\cos}^{2} \left(\theta\right) + {\sin}^{2} \left(\theta\right) = 1$

$\Rightarrow {\sin}^{2} \left(\theta\right) = 1 - {\left(- \frac{3}{5}\right)}^{2} = 1 - \frac{9}{25} = \frac{25}{25} - \frac{9}{25} = \frac{16}{25}$

Take the square root of both sides:

$\Rightarrow \sin \left(\theta\right) = \sqrt{\frac{16}{25}} = \frac{\sqrt{16}}{\sqrt{25}} = \pm \frac{4}{5}$

Now, is the answer the positive or negative square root? Well, you need to know the unit circle and it's symmetry properties. You will find that $\sin \left(\theta\right)$ is positive in the first and second quadrants. An angle between 90˚ and 180˚ corresponds to the second quadrant. Therefore, the angle is the positive root.

$\sin \left(\theta\right) = \frac{4}{5}$