Inverse Trigonometry Sheet 1

Principal Value Domains and Ranges

Principal Value Branches

The Range column represents the Principal Value Branch for each function.

Function Domain ($$x$$) Range (Principal Value)
$$\sin^{-1} x$$ $$[-1, 1]$$ $$[-\frac{\pi}{2}, \frac{\pi}{2}]$$
$$\cos^{-1} x$$ $$[-1, 1]$$ $$[0, \pi]$$
$$\tan^{-1} x$$ $$\mathbb{R}$$ (All Real Numbers) $$(-\frac{\pi}{2}, \frac{\pi}{2})$$
$$\text{cosec}^{-1} x$$ $$\mathbb{R} – (-1, 1)$$ $$[-\frac{\pi}{2}, \frac{\pi}{2}] – \{0\}$$
$$\sec^{-1} x$$ $$\mathbb{R} – (-1, 1)$$ $$[0, \pi] – \{\frac{\pi}{2}\}$$
$$\cot^{-1} x$$ $$\mathbb{R}$$ (All Real Numbers) $$(0, \pi)$$

Self-Adjusting Properties

Property Group 1

$$\sin(\sin^{-1} x) = x, \text{ if } x \in [-1, 1]$$
$$\sin^{-1}(\sin \theta) = \theta, \text{ if } \theta \in [-\frac{\pi}{2}, \frac{\pi}{2}]$$

Property Group 2

$$\cos(\cos^{-1} x) = x, \text{ if } x \in [-1, 1]$$
$$\cos^{-1}(\cos \theta) = \theta, \text{ if } \theta \in [0, \pi]$$

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