Inverse Trigonometry Sheet 3 (Updated)

Negative Arguments and All Reciprocal Relations

1. Negative Arguments ($$f^{-1}(-x)$$)

Property Domain Conditions
$$\sin^{-1}(-x) = -\sin^{-1} x$$ $$x \in [-1, 1]$$
$$\tan^{-1}(-x) = -\tan^{-1} x$$ $$x \in \mathbb{R}$$
$$\csc^{-1}(-x) = -\csc^{-1} x$$ $$|x| \geq 1$$
$$\cos^{-1}(-x) = \pi – \cos^{-1} x$$ $$x \in [-1, 1]$$
$$\sec^{-1}(-x) = \pi – \sec^{-1} x$$ $$|x| \geq 1$$
$$\cot^{-1}(-x) = \pi – \cot^{-1} x$$ $$x \in \mathbb{R}$$

2. All 6 Reciprocal Relations

Function Reciprocal Form Condition ($$x > 0$$)
$$\sin^{-1} x$$ $$\csc^{-1} \left(\frac{1}{x}\right)$$ $$x \in [-1, 1], x \neq 0$$
$$\csc^{-1} x$$ $$\sin^{-1} \left(\frac{1}{x}\right)$$ $$|x| \geq 1$$
$$\cos^{-1} x$$ $$\sec^{-1} \left(\frac{1}{x}\right)$$ $$x \in [-1, 1], x \neq 0$$
$$\sec^{-1} x$$ $$\cos^{-1} \left(\frac{1}{x}\right)$$ $$|x| \geq 1$$
$$\tan^{-1} x$$ $$\cot^{-1} \left(\frac{1}{x}\right)$$ $$x > 0$$
$$\cot^{-1} x$$ $$\tan^{-1} \left(\frac{1}{x}\right)$$ $$x > 0$$
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