AQA A Level Maths: Pure复习笔记5.5.4 Inverse Trig Functions

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 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°)

5.5.4-Inverse-Trig-Functs-Illustr-1_restr-sin

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

5.5.4-Inverse-Trig-Functs-Illustr-2_restr-cos

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

5.5.4-Inverse-Trig-Functs-Illustr-3_restr-tan

What does the graph of arcsin look like?

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

5.5.4-Inverse-Trig-Functs-Illustr-4_arcsin

  • 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 y = arccos x looks like this:

5.5.4-Inverse-Trig-Functs-Illustr-5_arccos5

  •  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 y = arctan x looks like this:

5.5.4-Inverse-Trig-Functs-Illustr-6_arctan

  • 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 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 y = 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

5.5.4-Inverse-Trig-Functs-Example

 

 

 

 

 

 

 

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