Question

If sin Θ = 2 over 3 and tan Θ < 0, what is the value of cos Θ?

Answer 1

The answer is -√5/3 or square root of five over three all negative.

I just took the test and this was correct!

Answer 2

$\tan\theta=\dfrac{\sin\theta}{\cos\theta}$

$\sin\theta=\dfrac23>0$, which tells us that $\cos\theta$ must be negative.

Recall that

$\sin^2\theta+\cos^2\theta=1\implies\cos\theta=-\sqrt{1-\sin^2\theta}$

where we take the negative square root because we know to expect $\cos\theta$. We find that

$\cos\theta=-\sqrt{1-\dfrac49}=-\dfrac{\sqrt5}3$