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

Dec 2, 2016

First of all, $\cot \theta = \frac{1}{\tan} \theta = \frac{1}{\frac{1}{4}} = 4$.

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 $\tan \theta = \text{opposite"/"adjacent}$, so our opposite side measures $- 4$ units and our adjacent side measures $- 1$ unit (because $\left(x , y\right) = \left(- , -\right)$ in quadrant 3).

We use pythagorean theorem to find the hypotenuse.

${\left(- 4\right)}^{2} + {\left(- 1\right)}^{2} = {h}^{2}$

$16 + 1 = {h}^{2}$

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:

$\sin \theta = \text{opposite"/"hypotenuse} = - \frac{4}{\sqrt{17}}$

$\csc \theta = \frac{1}{\sin} \theta = \text{hypotenuse"/"opposite} = - \frac{\sqrt{17}}{4}$

$\cos \theta = \text{adjacent"/"hypotenuse} = - \frac{1}{\sqrt{17}}$

$\sec \theta = \frac{1}{\cos} \theta = \text{hypotenuse"/"adjacent} = - \sqrt{17}$

Hopefully this helps!