Find this if $\mbox{cosec}( x) =$

Question

Given that $csc(x) = 9$

without a calculator evaluate:

i) $\cot(x)$

ii) $\tan(x)$

iii) $\cos(x)$

I know that $\csc(x) = \sin(x)$ divided by $1$. But I don't know what $x$ is. Not sure what to do to find $x$.

I know $\cot(x) = \tan(x)$ divided by $1$.

$\tan(x)$ is just the tan of $x$ (still need to find $x$ lol).

Lastly, I know that $\cos x$ is $\cos x$.

Answer 1

$$\csc(x)=9=\frac{1}{\sin(x)}=\frac{\text{hypotenuse}}{\text{adjacent}}$$

We don't need to know the exact length, but from this we know the ratios. We know that the sides of the hypotenuse and the adjacent can be, respectively, $9$ and $1$, $18$ and $2$, etc. Hence, we know that the third side can be $\sqrt{80}$, $\sqrt{320}=2\sqrt{80}$, etc. But basically, it won't matter because the $2$ or whatever the coefficient is will cancel and we will remain with the sides having the ratio of $\sqrt{80}:1:9$ (where $\sqrt{80}=4\sqrt{5}$ is the opposite, $1$ is the adjacent, and $9$ is, of course, the hypotenuse). This is all we care about for trig ratios.

With this, you should be able to answer everything.

i) $\cot(x)=\frac{1}{\tan(x)}=\frac{\text{adjacent}}{\text{opposite}}=\frac{1}{4\sqrt{5}}=\frac{\sqrt{5}}{20}$

ii) $\tan(x)=\frac{\text{opposite}}{\text{adjacent}}=\frac{4\sqrt{5}}{1}=4\sqrt{5}$

iii) $\cos(x)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{1}{9}$

Cheers! -Shahar

Answer 2

Suggest you draw a right triangle with hypotenuse 9 and vertical leg 1. Then $x$ is the angle opposite the 1.

(Check that $\csc x = 9$ for this triangle.)

Can you then find the length of the horizontal leg using Pythagorean theorem?

With all three legs known, computing the requested trigonometric ratios will be easy based on their definitions.