Question

How do you find the five remaining trigonometric function satisfying `tantheta=5/4`, `costheta<0`?

Answer 1

Find values of trig functions

Explanation:

Use trig identity:
`1 + tan^2 x = 1/(cos^2 x)`
`cos^2 x = 1/(1 + tan^2 x)`
`sin^2 x = 1/(1 + cot^2 x)`
In this case:
`cos^2 x = 1/(1 + 25/16) = 1/(41/16) = 16/41`
`cos x = - 4/sqrt41 = - (4sqrt41)/41`
Find sin x by the same way:
`sin^2 x = 1/(1 + 16/25) = 1/(41/25) = 25/41`
`sin x = - 5/(sqrt41)` (because `tan x = 5/4` > 0)
`tan x = sin/(cos) = 5/4`
`cot x = 1/(tan) = 4/5`
`sec x = 1/(cos) = - sqrt41/4`
`csc x = 1/(sin) = - sqrt41/5`