Question

How do you find the exact value of cos(theta) if sin(theta)=-2/3?

Answer 1

cos `theta`= `(sqrt5)/3`
or it could be cos `theta`= `(sqrt5)/-3`

Explanation:

Since sin `theta` is negative, it can be in the third or fourth quadrant

Drawing your right-angled triangle, place your `theta` in one of three corners. Your longest side will be 3 and the side opposite the `theta` will be -2. Finally, using Pythagoras theorem, your last side should be `sqrt5`

Now, if your triangle was in the third quadrant, you would have
cos `theta`= `(sqrt5)/-3` since cosine is negative in the third quadrant

But if your triangle was in the fourth quadrant, you would have
cos `theta`= `(sqrt5)/3` since cosine is positive in the fourth quadrant

Answer 2

`color(indigo)(cos theta = +- sqrt5 / 3`

Explanation:

`sin theta = -2/3`

`sin^2 theta = (-2/3)^2 = 4/9`

But `cos^2 theta = 1 - sin^2 theta = 1 - 4/9 = 5/9`

`:. cos theta = sqrt(5/9) = +- sqrt5 / 3`