8.3 Inverse trigonometric functions  (Page 6/15)

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Key concepts

• An inverse function is one that “undoes” another function. The domain of an inverse function is the range of the original function and the range of an inverse function is the domain of the original function.
• Because the trigonometric functions are not one-to-one on their natural domains, inverse trigonometric functions are defined for restricted domains.
• For any trigonometric function $f(x),$ if $x=f^{-1}(y),$ then $f(x)=y.$ However, $f(x)=y$ only implies $x=f^{-1}(y)$ if $x$ is in the restricted domain of $f.$ See [link] .
• Special angles are the outputs of inverse trigonometric functions for special input values; for example, $\frac{\pi }{4}={\tan }^{-1}(1)\text{and}\frac{\pi }{6}={\sin }^{-1}\left(\frac{1}{2}\right).$ See [link] .
• A calculator will return an angle within the restricted domain of the original trigonometric function. See [link] .
• Inverse functions allow us to find an angle when given two sides of a right triangle. See [link] .
• In function composition, if the inside function is an inverse trigonometric function, then there are exact expressions; for example, $\sin \left({\cos }^{-1}\left(x\right)\right)=\sqrt{1-x^2}.$ See [link] .
• If the inside function is a trigonometric function, then the only possible combinations are ${\sin }^{-1}\left(\cos x\right)=\frac{\pi }{2}-x$ if $0\le x\le \pi$ and ${\cos }^{-1}\left(\sin x\right)=\frac{\pi }{2}-x$ if $-\frac{\pi }{2}\le x\le \frac{\pi }{2}.$ See [link] and [link] .
• When evaluating the composition of a trigonometric function with an inverse trigonometric function, draw a reference triangle to assist in determining the ratio of sides that represents the output of the trigonometric function. See [link] .
• When evaluating the composition of a trigonometric function with an inverse trigonometric function, you may use trig identities to assist in determining the ratio of sides. See [link] .

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