# How do you find the values of the other five trigonometric functions of the acute angle A with cscA=3?

Mar 21, 2018

$\sin A = \frac{1}{3}$

$\cos A = \frac{2 \sqrt{2}}{3}$

$\tan A = \frac{1}{2 \sqrt{2}}$

$\csc A = \frac{3}{1}$

$\sec A = \frac{3}{2 \sqrt{2}}$

$\cot A = 2 \frac{\sqrt{2}}{1}$

#### Explanation:

$\csc A = 3$
$\csc A = \text{hypotenuse"/"opposite side} = \frac{3}{1}$

By pythagoras theorem

$\text{adjacent side} = \sqrt{{3}^{2} - {1}^{2}} = 2 \sqrt{2}$

We have

$\text{opposite side} = 1$
$\text{adjacent side} = 2 \sqrt{2}$
$\text{hypotenuse} = 3$

$\sin A = \text{opposite side"/"hypotenuse} = \frac{1}{3}$
$\cos A = \text{adjacent side"/"hypotenuse} = \frac{2 \sqrt{2}}{3}$
$\tan A = \text{opposite side"/"adjacent side} = \frac{1}{2 \sqrt{2}}$
$\csc A = \text{hypotenuse"/"opposite side} = \frac{3}{1}$
$\sec A = \text{hypotenuse"/"adjacent side} = \frac{3}{2 \sqrt{2}}$
$\cot A = \text{adjacent side"/"opposite side} = 2 \frac{\sqrt{2}}{1}$