Solve the trigonometry problem

Question

If cscΘ=3, Find cosΘ.

I am lost on what trig function to use and how would it be used since it is 3? Thank you!

Answer 1

Hi Christoforos,

Recall that

sin(θ) = 1 / csc(θ)

and

cos²(θ) + sin²(θ) = 1, so cos(θ) = ±√(1 - sin²(θ)).

Since csc(θ) = 3, you know that sin(θ) = 1/3, and sin²(θ) = 1/9. You can then use the above equation to figure out that cos²(θ) = 1 - 1/9 = 8/9, and cos(θ) = ±√(8/9). Simplifying this a bit further, we get that cos(θ) = ±2√(2) / 3. cos(θ) is positive if θ lies in quadrants I or IV, and negative if it lies in quadrants II or III, but since this is not specified in the problem, leave the ± in.

Hope this helps,

Akshat Y.

Answer 2

I made the comment about lengths not being negative, but in this case, we can consider that side to be negative since we are dealing with periodic functions over an indeterminate domain. So, include that plus or minus in the answer.