Question

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

Answer 1

`\sin \theta = \frac 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 √(1 - \sin ^2\theta)`

We know it's negative,

`\cos \theta = - √(1 - (2/3)^2) =-√(5/9) = - \frac 1 3 √(5)`

Answer: -(1/3)√5