Question

How do you find the other five trigonometric functions of x if `cosx = 3/5`?

Answer 1

It;s crucial to be familiar with the Pythagorean trigonometric identity and trigonometric identities corresponding to relations of sine and cosine with other trigonometric functions.

Explanation:

First we use the Pythagorean trigonometric identity:
`sin^2 x + cos^2 x =1`
`sin^2 x +(3/5)^2=1`
`sin^2 x +9/25=1`
`sin^2 x =1-9/25`
`sin^2 x=16/25`
`sin x=+-4/5`
Now, if we knew more about the angle `x` we could eliminate one of the possibilities for `sin x` but since we don't we should examine both options:

CASE 1: `sin x=4/5`
Using basic trigonometric identities:
`tan x=sin x/cos x=(4/5)/(3/5)=4/5*5/3=4/3`
`cot x=cosx/sinx=3/4`
`sec x=1/cos x=5/3`
`csc x=1/sin x=5/4`

CASE 2: `sin x=-4/5`
Using basic trigonometric identities:
`tan x=sin x/cos x=(-4/5)/(3/5)=-4/5*5/3=-4/3`
`cot x=cosx/sinx=-3/4`
`sec x=1/cos x=5/3`
`csc x=1/sin x=-5/4`