# Inverse Tangent Function

The inverse tangent function is denoted as $tan^{-1}(x)$. This function is used to find the unknown measure of an angle of a right triangle when two side lengths are known.The inverse of tangent is the inverse function of the tangent function and can be used to calculate angle measures when certain other things are known. This lesson will go into detail about the inverse tangent, give some formulas and follow up with a quiz to check your understanding.
The domain of the inverse of tangent function is (−∞,∞)and the range is $\frac{-\prod }{2},\frac{\prod }{2}$ . The inverse of the tangent function will yield values in the 1st and 4th quadrants.
$tan^{-1}(x)$ = arctan (x)

To evaluate inverse of tangent functions remember that the following statement is equivalent.
$\Theta = tan^{-1}(x) \Leftrightarrow x = tan(\Theta)$

In other words, when we evaluate tan inverse function we are asking what angle,θ, did we plug into the tangent function (regular, not inverse!) to get x.

 f(x)= tan(x) Domain $(\frac{-\prod }{2}< x < \frac{\prod}{2}$) Range (-∞ , ∞ ) $f(x)=tan^{-1}x$ (-∞ , ∞ ) $\frac{-\prod }{2}< y < \frac{\prod}{2}$

In tangent function, we know that $tan \Theta =\frac{opposite side }{Adjacent}$
The inverse of tangent function $tan{-1}$ takes the ratio opposite/adjacent and gives the angle θ.

∴ θ = $tan^{-1}(\frac{opposite side }{adjacent})$

For example, $tan 30^{0} =\frac{1}{\sqrt{3}}$

∴ θ = $tan^{-1}(\frac{1}{\sqrt{3}})$

⇒ θ = $30^{0}$

## Evaluate inverse tangent function

1) tan(x)= 300/400, find the value of x
Solution : tan(x)= 300/400
⇒ tan(x)= 0.75
∴ x = $tan^{-1}(0.75)$
⇒ x = $36.9^{0}$

2) Evaluate f(x) = $tan^{-1}(\frac{\sqrt{3}}{2})$

Solution : f(x) = $tan^{-1}(\frac{\sqrt{3}}{2})$

y = $tan^{-1}(\frac{\sqrt{3}}{2})$

⇒ y = $40.89^{0}$

3) Evaluate f(x) = $tan^{-1}(1)$

Solution : f(x) = $tan^{-1}(1)$

y = $tan^{-1}(1)$

⇒ y = $45^{0}$ ( For the degree you can use a unit circle or calculator)

precalculus

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