Question

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

Answer 1

`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`