# How do I determine the exact trigonometric value given CSC θ = √11/3, π/2 < θ < π; COT θ?

Feb 20, 2018

cot(θ)=-sqrt(2)/3

#### Explanation:

Recall this identity:

1+cot^2(θ)=csc^2(θ)

csc^2(θ)=11/9" " (assuming that the radical only comprises the $11$).

1+cot^2(θ)=11/9

cot^2(θ)=11/9 - 9/9 = 2/9

cot(θ)=+-sqrt(2/9)=+- sqrt(2)/3

We're between π/2 and π (the second quadrant). Here, cosine is negative and sine is positive; thus, cotangent (cosine over sine) is negative. The negative answer is what we're looking for.

cot(θ)=-sqrt(2)/3