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# Inverses by Mapping

## Reflect points from f(x) across the line y = x.

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Inverse of Trigonometric Ratios

REMEMBER: This connects with trigonometric functions so click here for a refresher on trig function and inverses.

The trigonometric inverse is a function that undoes a trig function to give the original argument of the function. It can also be used to find an angle from the ratio of two sides.

Caution: The trigonometric inverse is not the same as the sec, csc, and cotangent.

sinθ=yrsin1yr=θcosθ=xrcos1xr=θtanθ=yxtan1yx=θcotθ=xycot1xy=θcscθ=rycsc1ry=θsecθ=rxsec1rx=θ\begin{align*}\sin \theta = \frac{y}{r} \rightarrow \sin^{-1} \frac{y}{r} = \theta \cos \theta = \frac{x}{r} \rightarrow \cos^{-1} \frac{x}{r} = \theta\\ \tan \theta = \frac{y}{x} \rightarrow \tan^{-1} \frac{y}{x} = \theta \cot \theta = \frac{x}{y} \rightarrow \cot^{-1} \frac{x}{y} = \theta\\ \csc \theta = \frac{r}{y} \rightarrow \csc^{-1} \frac{r}{y} = \theta \sec \theta = \frac{r}{x} \rightarrow \sec^{-1} \frac{r}{x} = \theta\end{align*}

Exact Values for Inverse sine, cosine, and tangent

To find the exact valued you need to know the Unit Circle, click here

\begin{align*}\sin^{-1} \left ( -\frac{\sqrt{3}}{2} \right )\end{align*}

Solution: This is a value from the special right triangles and the unit circle.

Recall that \begin{align*}-\frac{\sqrt{3}}{2}\end{align*} is from the \begin{align*}30-60-90\end{align*} triangle. The reference angle for \begin{align*}\sin\end{align*} and \begin{align*}\frac{\sqrt{3}}{2}\end{align*} would be\begin{align*}60^\circ\end{align*} . Because this is sine and it is negative, it must be in the third or fourth quadrant. The answer is either \begin{align*}\frac{4\pi}{3}\end{align*} or \begin{align*}\frac{5\pi}{3}\end{align*} .

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#### Inverse of Functions Through Algebraic Manipulation

The horizontal line test is a test applied to a function to see how many times the graph of a function intersects an arbitrary horizontal line drawn across the coordinate grid. A function passes this test if it intersects the horizontal line in only one place, no matter where the horizontal line is drawn.

one to one function is a function that passes both the horizontal and vertical line tests.

The vertical line test is a test applied to a function to see how many times the graph of a function intersects an arbitrary vertical line drawn across the coordinate grid. A function passes this test if it intersects the vertical line in only one place, no matter where the vertical line is drawn.

What do you need to do the the regular function to find the inverse of it? Click here for guidence.

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#### Inverse by Mapping

Mapping is a procedure involving the plotting of points on a coordinate grid to see the behavior of a function.

The points \begin{align*}(a, b)\end{align*} and \begin{align*}(b, a)\end{align*} in the coordinate plane are symmetric with respect to the line \begin{align*}y = x\end{align*} .

The points \begin{align*}(a, b)\end{align*} and \begin{align*}(b, a)\end{align*} are reflections of each other across the line \begin{align*}y = x\end{align*} .

Remember: Not all inverses of a function will be functions.