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

Feb 20, 2018

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 ${180}^{\circ}$ and ${270}^{\circ}$, 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

$\tan \left(\theta\right) = \frac{\sqrt{8}}{8} = \frac{\sqrt{2}}{4}$