Inverse Trigonometric Functions (Edexcel IAL Maths)

Inverse Trig Functions

Spec point — 2.1

Inverse trigonometric functions

What are the inverse trigonometric functions?

• These functions are the inverse functions of sin, cos and tan

  • sin **(arcsin x) = x
  • cos**(arccos x) = x
  • tan (arctan x) = x

• The domains of sin, cos, and tan must first be restricted to make them one-to-one functions (only one-to-one functions have inverses)

What are the restricted domains?

• domain of sin x is restricted to-π/2 ≤ x ≤ π/2 (-90° ≤ x ≤ 90°)

• domain of cos x is restricted to 0 ≤ x≤ π (0° ≤ x ≤ 180°)

• domain of tan x is restricted to-π/2 < x < π/2 (-90° < x < 90°)

How do I sketch the graph of y = arcsin x?

• The graph of y= arcsin x looks like this:

• *the domain is -1 ≤ x ≤ 1
• the range is -π/2 ≤ arcsin x≤ π/2 (-90° ≤ arcsin x ≤ 90°)

How do I sketch the graph of y = arccos x?

• The graph of y= arccos x looks like this:

• *the domain is -1 ≤ x ≤ 1
• the range is 0 ≤ arccos x ≤ π(0° ≤ arccos x ≤ 180°)

How do I sketch the graph of y = arctan x?

• The graph of y= arctan x looks like this:

• the* domain is x ∈ ℝ (ie arctan x is defined for all real number values of x)
• the range is **-π/2 < arctan **x < π/2 (-90° < arctan x < 90°)
• horizontal asymptotes at y= - π/2** and y = π/2**

Exam Hint

• Make sure you know the shapes of the graphs for sin, cos and tan.
• As inverses, the graphs of arcsin, arccos and arctan are reflections of sin, cos and tan in the line **y **= x.
• The values returned by the sin^-1, cos^-1 and tan^-1keys on your calculator are the values from the ranges of arcsin, arccos and arctan.

Worked Example