Question

Given `cosalpha=1/3`, how do you find `sinalpha`?

Answer 1

`sin(a) = +-(2sqrt2)/3`

Explanation:

You can use a trig identity that states that `cos^2(a) = 1 - sin^2(a)`

so we get that

`1/9 = 1 - sin^2(a) => sin^2(a) = 8/9 => sin(a) = +-(2sqrt2)/3`

Answer 2

Use the identity `cos^2(alpha) + sin^2(alpha) = 1` and solve for `sin(alpha)`:

`sin(alpha) = +-sqrt(1 - cos^2(alpha))`

Explanation:

Substitute `(1/3)^2` for `cos^2(alpha)`

`sin(alpha) = +-sqrt(1 - (1/3)^2)`

`sin(alpha) = +-sqrt(1 - 1/9)`

`sin(alpha) = +-sqrt(8/9)`

`sin(alpha) = +-sqrt(8)/3`

`sin(alpha) = +-(2sqrt(2))/3`

Because we are not given any clue whether `alpha` is in the first or the fourth quadrant, then we cannot determine whether the sine function is positive or negative.