# How do you find inverse sin (-1/2)?

Oct 10, 2015

210°, 330°

#### Explanation:

Using your calculator, press "shift" "$\sin \left(\frac{1}{2}\right)$" and it will give you $30$. (Always ignore the negative when you are doing the inverse!)

But since this is a negative, you cannot just write 30° as your final answer.

Using the ASTC rule, you know that for $\sin$ to be positive it has to be in Quadrant 1 and 2. But since this is a negative it has to be the complete opposite! So Quadrant 3 and 4 is where $\sin$ will be negative.

In Quadrant 3, from the ASTC rule, take 180°+prop $\Rightarrow$ $\propto$ being the answer you just got aka 30°!

In Quadrant 4, from the ASTC rule, take 360°-prop .

So,

180°+30°=210°
360°-30°=330°

Oct 10, 2015

We use our knowledge of special angles together with the definition of inverse sine.

#### Explanation:

"inverse sine" may refer to either a single values function (the principal value -- common in introductory courses) or to a "multivalued function".

Here is the definition for the principal value:

$y = \arcsin x$ if and only if $\left(- \frac{\pi}{2} \le y \le \frac{\pi}{2} \boldsymbol{\text{ and }} \sin y = x\right)$

We know that $\sin \left(\frac{\pi}{6}\right) = \frac{1}{2}$ and so, $\sin \left(- \frac{\pi}{6}\right) = - \frac{1}{2}$.

Therefore, the (principal) inverse sine of $- \frac{1}{2}$ is $- \frac{\pi}{6}$.