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For any trigonometric function, $\text{\hspace{0.17em}}f\left({f}^{-1}\left(y\right)\right)=y\text{\hspace{0.17em}}$ for all $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ in the proper domain for the given function. This follows from the definition of the inverse and from the fact that the range of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ was defined to be identical to the domain of $\text{\hspace{0.17em}}{f}^{-1}.\text{\hspace{0.17em}}$ However, we have to be a little more careful with expressions of the form $\text{\hspace{0.17em}}{f}^{-1}\left(f\left(x\right)\right).$
Is it correct that $\text{\hspace{0.17em}}{\mathrm{sin}}^{-1}(\mathrm{sin}\text{\hspace{0.17em}}x)=x?$
No. This equation is correct if $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ belongs to the restricted domain $\text{\hspace{0.17em}}\left[-\frac{\pi}{2},\frac{\pi}{2}\right],\text{\hspace{0.17em}}$ but sine is defined for all real input values, and for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ outside the restricted interval, the equation is not correct because its inverse always returns a value in $\text{\hspace{0.17em}}\left[-\frac{\pi}{2},\frac{\pi}{2}\right].\text{\hspace{0.17em}}$ The situation is similar for cosine and tangent and their inverses. For example, $\text{\hspace{0.17em}}{\mathrm{sin}}^{-1}\left(\mathrm{sin}\left(\frac{3\pi}{4}\right)\right)=\frac{\pi}{4}.$
Given an expression of the form f ^{−1} (f(θ)) where $\text{\hspace{0.17em}}f(\theta )=\mathrm{sin}\text{\hspace{0.17em}}\theta ,\text{}\mathrm{cos}\text{\hspace{0.17em}}\theta ,\text{or}\mathrm{tan}\text{\hspace{0.17em}}\theta ,\text{\hspace{0.17em}}$ evaluate.
Evaluate the following:
Evaluate $\text{\hspace{0.17em}}{\mathrm{tan}}^{-1}\left(\mathrm{tan}\left(\frac{\pi}{8}\right)\right)\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}{\mathrm{tan}}^{-1}\left(\mathrm{tan}\left(\frac{11\pi}{9}\right)\right).$
$\frac{\pi}{8};\frac{2\pi}{9}$
Now that we can compose a trigonometric function with its inverse, we can explore how to evaluate a composition of a trigonometric function and the inverse of another trigonometric function. We will begin with compositions of the form $\text{\hspace{0.17em}}{f}^{-1}\left(g\left(x\right)\right).\text{\hspace{0.17em}}$ For special values of $\text{\hspace{0.17em}}x,$ we can exactly evaluate the inner function and then the outer, inverse function. However, we can find a more general approach by considering the relation between the two acute angles of a right triangle where one is $\text{\hspace{0.17em}}\theta ,\text{\hspace{0.17em}}$ making the other $\text{\hspace{0.17em}}\frac{\pi}{2}-\theta .$ Consider the sine and cosine of each angle of the right triangle in [link] .
Because $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta =\frac{b}{c}=\mathrm{sin}\left(\frac{\pi}{2}-\theta \right),\text{\hspace{0.17em}}$ we have $\text{\hspace{0.17em}}{\mathrm{sin}}^{-1}\left(\mathrm{cos}\text{\hspace{0.17em}}\theta \right)=\frac{\pi}{2}-\theta \text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}0\le \theta \le \pi .\text{\hspace{0.17em}}$ If $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ is not in this domain, then we need to find another angle that has the same cosine as $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ and does belong to the restricted domain; we then subtract this angle from $\text{\hspace{0.17em}}\frac{\pi}{2}.$ Similarly, $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta =\frac{a}{c}=\mathrm{cos}\left(\frac{\pi}{2}-\theta \right),\text{\hspace{0.17em}}$ so $\text{\hspace{0.17em}}{\mathrm{cos}}^{-1}\left(\mathrm{sin}\text{\hspace{0.17em}}\theta \right)=\frac{\pi}{2}-\theta \text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}-\frac{\pi}{2}\le \theta \le \frac{\pi}{2}.\text{\hspace{0.17em}}$ These are just the function-cofunction relationships presented in another way.
Given functions of the form $\text{\hspace{0.17em}}{\mathrm{sin}}^{-1}\left(\mathrm{cos}\text{\hspace{0.17em}}x\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{\mathrm{cos}}^{-1}\left(\mathrm{sin}\text{\hspace{0.17em}}x\right),\text{\hspace{0.17em}}$ evaluate them.
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