A-Level — mathsStandard1 min read

Inverse Trigonometric Functions

Work with arcsin, arccos, and arctan at A-Level. Understand domains, ranges, and graphs.

Tutor AI Team
Tutor AI Team

Inverse trig functions 'undo' the trig functions. Since trig functions are periodic, we restrict their domains to make them one-to-one.

The Functions

Function Domain Range
arcsin⁡x\arcsin x (sin⁡−1x\sin^{-1}x) [−1,1][-1, 1] [−π2,π2][-\frac{\pi}{2}, \frac{\pi}{2}]
arccos⁡x\arccos x (cos⁡−1x\cos^{-1}x) [−1,1][-1, 1] [0,π][0, \pi]
arctan⁡x\arctan x (tan⁡−1x\tan^{-1}x) (−∞,∞)(-\infty, \infty) (−π2,π2)(-\frac{\pi}{2}, \frac{\pi}{2})

Key Properties

  • arcsin⁡(sin⁡x)=x\arcsin(\sin x) = x only if x∈[−π2,π2]x \in [-\frac{\pi}{2}, \frac{\pi}{2}].
  • arcsin⁡x+arccos⁡x=π2\arcsin x + \arccos x = \frac{\pi}{2} for all x∈[−1,1]x \in [-1,1].

Graphs

Each is a reflection of the restricted trig function in y=xy = x.

Worked Examples

arcsin⁡(12)=π6\arcsin(\frac{1}{2}) = \frac{\pi}{6}. arccos⁡(0)=π2\arccos(0) = \frac{\pi}{2}. arctan⁡(1)=π4\arctan(1) = \frac{\pi}{4}.

Practice Problems

    1. Find arcsin⁡(−32)\arcsin(-\frac{\sqrt{3}}{2}).
    1. Find arccos⁡(−12)\arccos(-\frac{1}{2}).
    1. Sketch y=arctan⁡xy = \arctan x.

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Key Takeaways

  • ✓

    Inverse trig functions have restricted ranges.

  • ✓

    arcsin⁡+arccos⁡=π2\arcsin + \arccos = \frac{\pi}{2}.

  • ✓

    Values are given in radians at A-Level.

Tags:
#a-level#maths#pure#trigonometry#inverse-functions

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