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Pure · Trigonometry

Chapter 1 · 3

The idea

Inverse trigonometric functions

The functions arcsin, arccos and arctan — why the domains of sin, cos and tan must be restricted to invert them, the principal ranges, and evaluating exact values and composite expressions.

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Pure · Trigonometry

Inverse trigonometric functions

The functions arcsin, arccos and arctan — why the domains of sin, cos and tan must be restricted to invert them, the principal ranges, and evaluating exact values and composite expressions.

Why it works

The question with too many answers

Solve sin⁡θ=12\sin\theta = \tfrac12. You know θ=π6\theta = \tfrac{\pi}{6} works — but so does 5π6\tfrac{5\pi}{6}, and so does π6+2π\tfrac{\pi}{6} + 2\pi, and infinitely many more: the sine wave crosses height 12\tfrac12 again and again forever. Yet press sin⁡−1(0.5)\sin^{-1}(0.5) on a calculator and it answers π6\tfrac{\pi}{6} — one angle, the same one every time, no hesitation. How does it choose? That choice is exactly what arcsin⁡\arcsin, arccos⁡\arccos and arctan⁡\arctan are: each hands back one agreed-upon angle, called the principal value. This lesson is about the agreement.

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