Question

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

Answer 1

$\sin \theta = \dfrac 2 3$

$\tan \theta < 0$

Positive sine, negative tangent, means we have a negative cosine.  We're talking about the second quadrant.

$\cos^2 \theta + \sin ^2 \theta = 1$

$\cos^2 \theta = 1 - \sin ^2\theta$

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

We know it's negative,

$\cos \theta = - \sqrt{1 - (2/3)^2} =-\sqrt{5/9} = - \dfrac 1 3 \sqrt{5}$

Answer: -(1/3)√5