# How do you simplify sin[Tan^-1 (-2/5) + Cos^-1 (3/5)]?

Jul 2, 2016

$\frac{14}{5 \sqrt{29}}$, against principal values of inverse functions, in the expression. General values are $\pm \frac{14}{5 \sqrt{29}} \mathmr{and} \pm \frac{26}{5 \sqrt{29}}$.

#### Explanation:

Let $a = {\tan}^{- 1} \left(- \frac{2}{5}\right)$. Then, $\tan a = - \frac{2}{5} < 0$.

So, principal a is in the 4th quadrant.

And so, sin is negative and cos is positive.

Thus, in this case, $\cos a = \frac{5}{\sqrt{29}} \mathmr{and} \sin a = - \frac{2}{\sqrt{29}}$.

Let $b = {\cos}^{- 1} \left(\frac{3}{5}\right)$. Then, #cos b =3/5 > 0.

So, principal b is in the 1st quadrant.

And so, sin is positive..

Thus, in this case, sin b = 4/5.

The given expression = sin (a + b)=sin a cos b+cos a sin b

$= \left(- \frac{2}{\sqrt{29}}\right) \left(\frac{3}{5}\right) + \left(\frac{5}{\sqrt{29}}\right) \left(\frac{4}{5}\right)$

$= \frac{14}{5 \sqrt{29}}$.

In general, a is in either 4th quadrant or in the 2nd and b is in either

1st or the 2nd. So, sin a and cos a can take both signs.

ln general, b is in either 1st or 4th. So, sin b can take both signs.

Accordingly, the general values are

$\pm \frac{14}{5 \sqrt{29}} \mathmr{and} \pm \frac{26}{5 \sqrt{20}}$.