Question

How do you find the remaining trigonometric ratios if `tan(alpha) = 9` and `0 < alpha < pi/2`?

Answer 1

Use SOH-CAH-TOA mnemonic and Pythagoras to find:

`sin(alpha) = 9/sqrt(82)` and `cos(alpha) = 1/sqrt(82)`

`csc(alpha) = sqrt(82)/9`, `sec(alpha) = sqrt(82)` and `cot(alpha) = 1/9`

Explanation:

`tan(alpha) = 9` is the `"opposite"/"adjacent"` ratio of a right angled triangle with angle `alpha`.

For our purposes, it does not matter what size the triangle is - just its proportions. So let the length of the opposite side be `9` and the length of the adjacent side be `1`. Then the length of the hypotenuse is `sqrt(9^2+1^2) = sqrt(82)`.

Hence `sin(alpha) = "opposite"/"hypotenuse" = 9/sqrt(82)`

and `cos(alpha) = "adjacent"/"hypotenuse" = 1/sqrt(82)`

Answer 2

Find other trig functions knowing tan x = 9

Explanation:

Use a calculator.
tan x = 9 --> x = 83.66 deg
sin 83.66 = 0.99
cos 83.66 = 0.11
`cot x = 1/9`
`sec x = 1/0.11 = 9.09`
`csc x = 1/0.99 = 1.01`