Question

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

Answer 1

`210°, 330°`

Explanation:

Using your calculator, press "shift" "`sin (1/2)`" 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` `rArr` `prop` 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°`

Answer 2

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=arcsinx` if and only if `(-pi/2 <= y <= pi/2 bb" and " siny=x)`

We know that `sin(pi/6) = 1/2` and so, `sin(-pi/6) = -1/2`.

Therefore, the (principal) inverse sine of `-1/2` is `-pi/6`.