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
- If sin x = 0.5 then arcsin 0.5 = x for values of x between -π/2 ≤ x ≤ π/2
- 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
Exam Tip
- Make sure you know the shapes of the graphs for sin, cos and tan so that you can easily reflect them in the line and hence sketch the graphs of arcsin, arccos and arctan
Worked Example
Given that satisfies the equation where , state the range of possible values of .