Question

How do you find the exact value for `sin (cos^-1 (-1/4))`?

Answer 1

`cos^-1(-1/4)` is a `t` in `[0, pi]` with `cos t = -1/4`

Since `t` is in quadrant 1 or 2, `sint` is positive.

The problem becomes: `cost=-1/4` and `sint >0`. Find `sint`.

Either draw a picture of `t` on the coordinate system,
or draw a right triangle with acute angle `t`, adjacent side 1 and hypotenuse 4
or use `sin^2t+cos^2 = 1` to get

`sint = sqrt15/4`