Question

How do you find the values of the trigonometric functions of θ from the information given cot θ = 1/4, sin θ < 0?

Answer 1

First of all, `cottheta = 1/tantheta = 1/(1/4) = 4`.

The problem says that sine is negative and we can see above that tangent is negative. Using the C-A-S-T sign rule, we determine that `theta` is in Quadrant 3.

We know that `tantheta = "opposite"/"adjacent"`, so our opposite side measures `-4` units and our adjacent side measures `-1` unit (because `(x, y) = (-, -)` in quadrant 3).

We use pythagorean theorem to find the hypotenuse.

`(-4)^2 + (-1)^2 = h^2`

`16 + 1 =h^2`

`h =+-sqrt( 17`

However, the hypotenuse can never be negative, so we only keep `sqrt(17)`.

We can now fill in all four of the other ratios:

`sintheta = "opposite"/"hypotenuse" = -4/sqrt(17)`

`csctheta = 1/sintheta = "hypotenuse"/"opposite" = -sqrt(17)/4`

`costheta = "adjacent"/"hypotenuse" = -1/sqrt(17)`

`sectheta = 1/costheta = "hypotenuse"/"adjacent" = -sqrt(17)`

Hopefully this helps!