Inverse Trigonometric Functions

Introduction

If sin x = 1/2, we can write one value of x = π/6.

If sin x = 1/3, i.e. x is not a well known angle, then we can write x = sin^-1 1/3.

Similarly,

cos x = t ⇒ x = cos^-1 t.

tan x = t ⇒ x = tan^-1 t.

Rules for defined values of sin^-1 x, cos^-1 x

1. y = sin^-1 x:
   Domain: x Є [-1, 1]
   Range (principal value branch of sin^-1 x)
   y Є [-π/2, π/2]

1. y = cos^-1 x:
   Domain: x Є [-1, 1]
   Range (principal value branch of cos^-1 x)
   y Є [0, π]

1. y = tan^-1 x:
   Domain: x Є R
   Range (principal value branch of tan^-1 x)
   y Є (-π/2, π/2)

1. y = cosec^-1 x:
   Domain: x Є (-∞, -1] ∪ [1, ∞)
   Range (principal value branch of cosec^-1 x)
   y Є [-π/2, 0) ∪ (0, π/2)

1. y = sec^-1 x:
   Domain: x Є (-∞, -1] ∪ [l, ∞)
   Range (principal value branch of sec^-1 x)
   y Є [0, π/2) ∪ (π/2, π]

1. y = cot^-1 x:
   Domain: X Є R
   Range (principal value branch of cot^-1 x)
   y Є (0, π)

Note the similarity in principal value branch of sin^-1 x, cosec^-1 x, tan^-1 x.

Interval for allowed values of y is known as principal value branch of that inverse function.

Important Results

Important Results (I):

1. sin (sin^-1x) = x, cos (cos^-1x) = x, ......
2. sin^-1 sin θ = θ, cos^-1 cos θ = θ
   if θ allows the restrictions on y in the definition of corresponding inverse function.
   e.g. sin^-1 sin2π/3 2π/3 because 2π/3 does not lie in the principal value branch of sin^-1 x.
   Hence sin^-1 sin2π/3 = sin^-1 sin(π - π/3)sin^-1 sinπ/3 = π/3.
3. sin^-1(-x) = -sin^-1x cos^-1(-x) = π - cos^-1x
   cosec^-1(-x) = -cosec^-1x sec^-1(-x) = π - sec^-1x
   tan^-1(-x) = -tan^-1x cot^-1(-x) = π - cot^-1x

Important Results (II):

1. If x > 0, y > 0 then
   
2. lf x > 0, y > 0 then
   
3. If x > 0
   
4. 

Important Results (III):

Results (IV):

sin^.-1 x + sin^.-1 y= sin^.-1 , when x ≥ 0, y ≥ 0, x² + y² ≤ 1

sin^.-1 x + sin^.-1 y= π - sin^.-1 , when x ≥ 0, y ≥ 0, x² + y² > 1

Results (V):

sin^.-1 x - sin^.-1 y= sin^.-1 , 0 ≤ y ≤ x

Results (VI):

cos^.-1 x + cos^.-1 y = cos^.-1 , where x ≥ 0, y ≥ 0

cos^.-1 x - cos^.-1 y = cos^.-1 , where 0 ≤ x, ≤ y