Question

Find sin x/2, cos x/2, and tan x/2 from the given information. cot x=5, 180°

Answer 1

The exact values of the trigonometric functions are: sin 0.5θ = 0.995, cos 0.5θ = 0.099, tan 0.5θ = 10.051.

How to determine the exact values of trigonometric functions

Herein we must determine the exact values of trigonometric functions based on definitions of trigonometric functions and trigonometric formulae. Now we introduce all expressions needed:

`\cot x = (x)/(y)`

`\tan \theta =(y)/(x)`

`\sin \theta = (y)/(√(x^2 + y^2))`

`\cos \theta = (x)/(√(x^2 + y^2))`

`\sec \theta = (√(x^2 + y^2))/(y)`

`\csc \theta = (√(x^2 + y^2))/(x)`

`\tan 0.5\theta = (\sin 0.5\theta)/(\cos 0.5\theta) = (\tan \theta)/(1 + \sec \theta)`

`\sin 0.5\theta = \pm \sqrt{(1 - \cos \theta)/(2) }`

`\cos 0.5\theta = \pm \sqrt{(1 + \cos \theta)/(2)}`

Based on statement, x, y < 0:

x = - 5, y = - 1

`\cos \theta = (- 5)/(√((- 5)^2 + (- 1)^2))`

`\cos \theta = - (5√(26))/(26)`

`\sin 0.5\theta = \sqrt{(1 - \left(-(5√(26))/(26) \right))/(2) }`

sin 0.5θ = 0.995

`\cos 0.5\theta = \sqrt{(1 + \left(-(5√(26))/(26) \right))/(2) }`

cos 0.5θ = 0.099

`\tan 0.5\theta = (0.995)/(0.099)`

tan 0.5θ = 10.051

The trigonometric functions has the following exact values: sin 0.5θ = 0.995, cos 0.5θ = 0.099, tan 0.5θ = 10.051.