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

1 Answer
Aug 8, 2015

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