Question

How do you find the exact values of `costheta` and `tantheta` when `sintheta=-1/2`?

Answer 1

`cos theta = pm sqrt{3}/2`

`tan theta = pm 1/sqrt{3}`

Explanation:

Get a new triangle, question writer! Seriously, there's nothing about this question that requires using one of the two cliche triangles of trig, yet this question does.

`cos ^2 theta + sin ^2 theta = 1`

`cos ^2 theta = 1 - sin ^2 theta`

`cos theta = pm \sqrt{ 1 - sin ^2 theta }`

`cos theta = pm sqrt{ 1 - (-1/2)^2} = pm sqrt{3}/2`

`tan theta = {sin theta}/{cos theta} = {-1/2}/{pm sqrt{3}/2}= pm 1/sqrt{3}`

Knowing only the sine, we cannot determine the sign of the cosine or of the tangent.