Question

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

Answer 1

Sinθ = (2/3)    tanθ < 0  Implies tanθ = Negative.

For sin positive and tan negative, implies θ is in the second quadrant.

Sinθ = (2/3) = opposite / Hypotenuse.

Adjacent side:

x² + 2² = 3²

x² + 4 = 9

x² = 9 - 4

x² = 5

x = √5 = Adjacent.

cosθ  =  Adjacent / Hypotenuse =  √5 / 3

Since θ is in the second quadrant, cosine is also negative in the second quadrant.


Therefore:

cosθ  =  - (√5/3)

The last option from the top :  negative square root of 5 over 3