Question

How do you find the values of the trigonometric functions of $\theta$ from the information given $cot\theta =\frac{1}{4},\sin \theta <0$?

Answer 1

Step 1. Find the values of the trigonometric functions of $\theta$.

Consider the given information $cot\theta =\frac{1}{4},\sin \theta <0$.

To begin with,

$\begin{array}{rcl}\tan \theta & =& \frac{1}{cot\theta }\\ & =& \frac{1}{{\displaystyle \frac{1}{4}}}\\ & =& 4\end{array}$

According to the issue, sine is negative, and as we can see above, the tangent is also negative.

Step 2: Apply the C-A-S-T sign rule:

Using the C-A-S-T sign rule, we determine that $\theta$ is in Quadrant $3$.

We know that,

$\tan \theta =\frac{\mathrm{opposite}}{\mathrm{adjacent}}$,

So,

The opposite side measures is $-4$ units and the adjacent side measures is $-1$ unit (As a result of $\left(x,y\right)=\left(-,-\right)$ in quadrant $3$).

The hypotenuse is located using the Pythagorean Theorem:

$\begin{array}{rcl}{(-4)}^2+{(-1)}^2& =& h^2\\ & \Rightarrow & 16+1=h^2\\ & \Rightarrow & h=\pm \sqrt{17}\end{array}$

However, the hypotenuse can never be negative. So $\sqrt{17}$.

All four of the remaining ratios can now be calculated as:

$\begin{array}{rcl}\sin \theta & =& \frac{\mathrm{opposite}}{\mathrm{hypotenuse}}\\ & =& -\frac{4}{\sqrt{17}}\end{array}$

And,

$\begin{array}{rcl}cosec\theta & =& \frac{1}{\sin \theta }\\ & =& \frac{\mathrm{hypotenuse}}{\mathrm{opposite}}\\ & =& -\frac{\sqrt{17}}{4}\end{array}$

And,

$\begin{array}{rcl}\cos \theta & =& \frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}\\ & =& -\frac{1}{\sqrt{17}}\end{array}$

And,

$\begin{array}{rcl}sec\theta & =& \frac{1}{\cos \theta }\\ & =& \frac{\mathrm{hypotenuse}}{\mathrm{adjacent}}\\ & =& -\sqrt{17}\end{array}$

Hence, the values of the trigonometric functions of $\theta$ from the information given $cot\theta =\frac{1}{4},\sin \theta <0$ are calculated above.