Inverse Trig Functions (Edexcel International A Level Maths: Pure 3)

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Inverse Trig Functions

What are arcsin, arccos and arctan?

  • 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 sincos, 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°)

Inverse Trig Functs Illustr 1_restr sin, A Level & AS Maths: Pure revision notes

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

Inverse Trig Functs Illustr 2_restr cos, A Level & AS Maths: Pure revision notes

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

 Inverse Trig Functs Illustr 3_restr tan, A Level & AS Maths: Pure revision notes

What does the graph of arcsin look like?

  • The graph of = arcsin x looks like this:

Inverse Trig Functs Illustr 4_arcsin, A Level & AS Maths: Pure revision notes

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

What does the graph of arccos look like?

  • The graph of = arccos x looks like this:

Inverse Trig Functs Illustr 5_arccos5, A Level & AS Maths: Pure revision notes

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

What does the graph of arctan look like?

  • The graph of = arctan x looks like this:

 

Inverse Trig Functs Illustr 6_arctan, A Level & AS Maths: Pure revision notes

  • the domain is   (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

Examiner Tip

  • 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 = x.
  • The values returned by the sin-1, cos-1 and tan-1 keys on your calculator are the values from the ranges of arcsin, arccos and arctan.

Worked example

Inverse Trig Functs Example, A Level & AS Maths: Pure revision notes

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Roger

Author: Roger

Expertise: Maths

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.