Question

If sin theta = 2/3 and tan theta <0 what is the value of cos theta?

a) (sqrt5)/2
b) -sqrt5
c) (sqrt5)/3
d) -(sqrt5)/3

Answer 1

d)
`\cos(\theta)=-(√(5))/(3)`

Explanation:

If
`\sin(\theta)=(2)/(3)` and
`\tan(\theta)\:<\:0`, then

`\theta` is in quadrant 2.

Recall that;

`\sin^2(\theta)+\cos^2(\theta)=1`

We substitute the given sine ratio to obtain;

`((2)/(3))^2+\cos^2(\theta)=1`

`(4)/(9)+\cos^2(\theta)=1`

`\cos^2(\theta)=1-(4)/(9)`

`\cos^2(\theta)=(5)/(9)`

`\cos(\theta)=\pm \sqrt{(5)/(9)}`

`\cos(\theta)=\pm (√(5))/(3)`

We are in the second quadrant, therefore

`\cos(\theta)=-(√(5))/(3)`