Question

How do I determine the exact trigonometric value given SIN θ = -1/3, 180 < θ < 270; TAN θ?

Answer 1

`tan(θ)=1/sqrt(8)`

Explanation:

Recall this identity

`sin^2(θ)+cos^2(θ)=1`

`sin^2(θ)=(-1/3)^2=1/9`

`1/9+cos^2(θ)=1`

`cos^2(θ)=9/9 - 1/9 = 8/9`

`cos(θ) = +-sqrt(8/9)=+- sqrt(8) / 3`

When we're between `180^@` and `270^@`, we're in the third quadrant, where sine and cosine are both negative. This means that we want the negative answer.

`cos(θ)=- sqrt(8) / 3`

So

`tan(θ)=sin(θ)/cos(θ)`

`tan(θ)=(-1/3) / (-sqrt(8) / 3)`

`tan(θ)=1/sqrt(8)" "` (negatives cancel out on division)

or

`tan(theta) = sqrt(8)/8 = sqrt(2)/4`