3.7.2 Inverse Trig Functions

What are the inverse trig functions?

• The functions arcsin, arccos and arctan are the inverse functions of sin, cos and tan respectively when their domains are restricted
• sin (arcsin x) = x  for  -1 ≤ x ≤ 1
• cos (arccos x) = x  for  -1 ≤ x ≤ 1
• tan (arctan x) = x  for all x
• You will have seen and used the inverse trig operations many times already
• Arcsin is the operation sin^-1  
• Arccos is to the operation cos^-1 
• Arctan is the operation tan^-1
• The domains of sin, cos, and tan must first be restricted to make them one-to-one functions
• A function can only have an inverse if it is a one-to-one function
• The domain of sin x is restricted to -π/2 ≤ x ≤ π/2  (-90° ≤ x ≤ 90°)
• The domain of cos x is restricted to 0 ≤ x ≤ π  (0° ≤ x ≤ 180°)
• The domain of tan x is restricted to -π/2 < x < π/2  (-90° < x < 90°)
• Be aware that sin^-1 x, cos^-1 x, and tan^-1 x are not the same as the reciprocal trig functions
• They are used to solve trig equations such as sin x = 0.5 for all values of x
• arcsin x is the same as sin^-1 x but not the same as (sin x)^-1  

What do the graphs of the inverse trig functions look like?

• The graphs of arcsin, arccos and arctan are the reflections of the graphs of sin, cos and tan (after their domains have been restricted) in the line y = x
• The domains of arcsin x and arccos x are both -1 ≤ x ≤ 1
• The range of arcsin x is -π/2 ≤ y ≤ π/2

• The range of arccos x is 0 ≤ y ≤ π

• The domain of arctan x is x ∈ ℝ
• The range of arctan x is -π/2 < y < π/2
• Note that there are horizontal asymptotes at π/2 and -π/2

How are the inverse trig functions used?

• The functions arcsin, arccos and arctan are used to evaluate trigonometric equations such as sin x = 0.5
  • If sin x = 0.5 then arcsin 0.5 = x for values of x between -π/2 ≤ x ≤ π/2
    • You can then use symmetries of the trig function to find solutions over other intervals
• The inverse trig functions are also used to help evaluate algebraic expressions 
  • From sin (arcsin x) = x we can also say that sinⁿ(arcsin x) = xⁿ   for  -1 ≤ x ≤ 1
  • If using an inverse trig function to evaluate an algebraic expression then remember to consider the domain and range of the function
    • arcsin(sin x) = x  only for  -π/2 ≤ x ≤ π/2
    • arccos(cos x) = x  only for  0 ≤ x ≤ π
    • arctan(tan x) = x  only for  -π/2 < x < π/2
  • The symmetries of the trig functions can be used when values lie outside of the domain or range
    • Using sin(x) = sin(π - x) you get arcsin(sin(2π/3)) = arcsin(sin(π/3)) = π/3