# How do you evaluate sin(arctan(3/4)) without a calculator?

Sep 11, 2016

$\sin \left(\arctan \left(\frac{3}{4}\right)\right) = \pm \frac{3}{5}$

#### Explanation:

$\sin \left(\arctan \left(\frac{3}{4}\right)\right)$ literally means sine of an angle whose tangent is $\frac{3}{4}$. Let the angle be $\theta = \arctan \left(\frac{3}{4}\right)$

As $\tan \theta = \frac{3}{4}$, $\cot \theta = \frac{4}{3}$ (as it is reciprocal)

Hence $\csc \cdot 2 \theta = 1 + {\cot}^{2} \theta = 1 + \frac{16}{9} = \frac{25}{9}$

Hence $\csc \theta = \pm \frac{5}{3}$ and $\sin \theta = \pm \frac{3}{5}$

Note that as $\tan \theta$ is positive, it is in firs or third quadrant and hence, we can have $\sin \theta$ positive as well as negative.