Inverse Trig Functions

If you restrict $f(x) = \sin x$ to the interval $-\dfrac{\pi}{2} \le x \le \dfrac{\pi}{2}$ , the function increases:

$$\hbox{\epsfysize=1.75in \epsffile{invtrig1.eps}}$$

This implies that the function is one-to-one, and hence it has an inverse. The inverse is called the inverse sine or arcsine function, and is denoted $\arcsin x$ or $\sin^{-1}(x)$ . Note that in the second case $\sin^{-1}(x)$ does not mean "$\dfrac{1}{\sin x}$ "!

Thus, $y = \sin^{-1} x$ is the angle whose sine is x. Another way of saying this is:

$$y = \sin^{-1} x \quad\hbox{is the same as}\quad \sin y = x.$$

The fact that $\sin$ and $\sin^{-1}$ are inverse functions can be expressed by the following equations:

$$\sin \sin^{-1} a = a \quad\hbox{for}\quad -1 \le a \le 1,$$

$$\sin^{-1} \sin b = b \quad\hbox{for}\quad -\dfrac{\pi}{2} \le b \le \dfrac{\pi}{2}.$$

Since the restricted $\sin$ takes angles in the range $-\dfrac{\pi}{2} \le x \le \dfrac{\pi}{2}$ and produces numbers in the range $-1 \le y \le 1$ , $\sin^{-1}$ takes numbers in the range $-1 \le y \le 1$ and produces angles in the range $-\dfrac{\pi}{2} \le x \le \dfrac{\pi}{2}$ .

$$\hbox{\epsfysize=2in \epsffile{invtrig2.eps}}$$


Example.

$$\arcsin \dfrac{1}{2} = \dfrac{\pi}{6}, \quad\hbox{since}\quad \sin \dfrac{\pi}{6} = \dfrac{1}{2}.$$

$$\sin^{-1} (-1) = -\dfrac{\pi}{2}, \quad\hbox{since}\quad \sin \left(-\dfrac{\pi}{2}\right) = -1.$$

Sine and arcsine are inverses, so they undo one another --- but you have to be careful!

$$\sin \left(\arcsin \dfrac{2}{5}\right) = \dfrac{2}{5}, \quad\hbox{but}\quad \arcsin \left(\sin 2\pi\right) = 0, \quad\hbox{not}\quad 2\pi.$$

$\arcsin (\hbox{stuff})$ can't be $2\pi$ , because $\arcsin$ always returns an angle in the range $-\dfrac{\pi}{2} \le x \le \dfrac{\pi}{2}$ .


Example. Find $\tan \sin^{-1} \dfrac{5}{13}$ .

First, let $\theta = \sin^{-1}
   \dfrac{5}{13}$ . This means that $\sin \theta = \dfrac{5}{13}$ . Now $\sin
   \theta = \dfrac{\rm opposite}{\rm hypotenuse}$ , so I get the following picture:

$$\hbox{\epsfysize=1.5in \epsffile{invtrig3.eps}}$$

I got the adjacent side using Pythagoras: $\sqrt{13^2 - 5^2} = 12$ .

Using the triangle, I have

$$\tan \sin^{-1} \dfrac{5}{13} = \tan \theta = \dfrac{5}{12}. \quad\halmos$$


You can find a derivative formula for $\arcsin$ using implicit differentiation. Let $y = \arcsin x$ . This is equivalent to $x = \sin y$ . Differentiate implicitly:

$$x = \sin y, \quad 1 = (\cos y)y', \quad y' = \dfrac{1}{\cos y}.$$

I'd like to express the result in terms of x. Here's the right triangle that says $x = \sin y$ :

$$\hbox{\epsfysize=1.5in \epsffile{invtrig4.eps}}$$

I found the other leg using Pythagoras. You can see that $\cos y = \sqrt{1 - x^2}$ . Hence, $y'
   = \dfrac{1}{\sqrt{1 - x^2}}$ . That is,

$$\der {} x \arcsin x = \dfrac{1}{\sqrt{1 - x^2}}.$$

Every derivative formula gives rise to a corresponding antiderivative formula:

$$\int \dfrac{1}{\sqrt{1 - x^2}}\,dx = \arcsin x + C.$$

Before I do some calculus examples, I want to mention some of the other inverse trig functions. I'll discuss the inverse cosine, inverse tangent, and inverse secant functions.

$$\hbox{\epsfysize=2in \epsffile{invtrig5.eps}}$$

$$\hbox{\epsfysize=2in \epsffile{invtrig6.eps}}$$

$$\hbox{\epsfysize=2in \epsffile{invtrig7.eps}}$$

As with $\sin$ and $\sin^{-1}$ , the domains and ranges of these functions and their inverses are "swapped":

$$\vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & Function & & Domain & & Range & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & $\arcsin$ & & $-1 \le x \le 1$ & & $-\dfrac{\pi}{2} \le x \le \dfrac{\pi}{2}$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & $\arccos$ & & $-1 \le x \le 1$ & & $0 \le x \le \pi$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & $\arctan$ & & $-\infty < x < \infty$ & & $-\dfrac{\pi}{2} < x < \dfrac{\pi}{2}$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & $\arcsec$ & & $x \le -1, x \ge 1$ & & $0 \le x < \dfrac{\pi}{2}, \dfrac{\pi}{2} < x \le \pi$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} }} $$


Example.

$$\tan^{-1} 1 = \dfrac{\pi}{4}, \quad\hbox{since}\quad \tan \dfrac{\pi}{4} = 1.$$

$$\cos^{-1} \left(-\dfrac{1}{2}\right) = \dfrac{2\pi}{3}, \quad\hbox{since}\quad \cos \dfrac{2\pi}{3} = -\dfrac{1}{2}. \quad\halmos$$


You can derive the derivative formulas for the other inverse trig functions using implicit differentiation, just as I did for the inverse sine function.

$$\der {} x \arccos x = -\dfrac{1}{\sqrt{1 - x^2}}$$

$$\der {} x \arctan x = \dfrac{1}{1 + x^2}$$

$$\der {} x \arcsec x = \dfrac{1}{|x|\sqrt{x^2 - 1}}$$


Example. Derive the formula for $\der {} x \sec^{-1} x$ .

The derivation starts out like the derivation for $\der {} x \sin^{-1} x$ . Let $y = \sec^{-1}
   x$ , so $\sec y = x$ . Differentiating implicitly, I get

$$(\sec y \tan y)y' = 1, \quad\hbox{so}\quad y' = \dfrac{1}{\sec y \tan y}.$$

There are two cases, depending on whether $x \ge 1$ or $x \le -1$ .

$$\hbox{\epsfysize=1.5in \epsffile{invtrig8.eps}}$$

Suppose $x \ge 1$ . Then $y = \sec^{-1}
   x$ is in the interval $0 \le y < \dfrac{\pi}{2}$ , as illustrated in the first diagram above. You can see from the picture that

$$\sec y = x \quad\hbox{and}\quad \tan y = \sqrt{x^2 - 1}.$$

Therefore,

$$y' = \dfrac{1}{\sec y \tan y} = \dfrac{1}{x\sqrt{x^2 - 1}}.$$

$x \ge 1$ , so x is positive, and $x = |x|$ . Therefore,

$$y' = \dfrac{1}{x\sqrt{x^2 - 1}} = \dfrac{1}{|x|\sqrt{x^2 - 1}}.$$

Now suppose that $x \le -1$ . Then $y =
   \sec^{-1} x$ is in the interval $\dfrac{\pi}{2} < y \le \pi$ , as illustrated in the second diagram above. Since x is negative, the hypotenuse must be $-x$ , since it must be positive and since $\sec y = \dfrac{(\hbox{hypotenuse})}{(\hbox{adjacent})}$ must equal x. In this case,

$$\sec y = x \quad\hbox{and}\quad \tan y = -\sqrt{x^2 - 1}.$$

Therefore,

$$y' = \dfrac{1}{\sec y \tan y} = \dfrac{1}{-x\sqrt{x^2 - 1}}.$$

$x \le -1$ , so x is negative, and $-x = |x|$ . Therefore,

$$y' = \dfrac{1}{-x\sqrt{x^2 - 1}} = \dfrac{1}{|x|\sqrt{x^2 - 1}}.$$

This proves that $y' =
   \dfrac{1}{|x|\sqrt{x^2 - 1}}$ in all cases.


Example.

$$\der {} x \left(\arcsin \sqrt{x} + \sqrt{\arcsin x}\right) = \dfrac{1}{\sqrt{1 - x}}\cdot \dfrac{1}{2\sqrt{x}} + \dfrac{1}{2}\left(\arcsin x\right)^{-1/2}\cdot \dfrac{1}{\sqrt{1 - x^2}}.$$

$$\der {} x \dfrac{1}{\tan^{-1} x} = \left(-\dfrac{1}{(\tan^{-1} x)^2}\right) \left(\dfrac{1}{1 + x^2}\right).$$

$$\der {} x \arcsec (e^x) = \dfrac{e^x}{e^x\sqrt{e^{2x} - 1}} = \dfrac{1}{\sqrt{e^{2x} - 1}}.$$

I don't need absolute values in the last example, because $e^x$ is always positive.


Example.

$$\der {} x \arctan \dfrac{1}{w} = \dfrac{-\dfrac{1}{w^2}}{1 + \dfrac{1}{w^2}} = -\dfrac{1}{1 + w^2}.$$

Hence,

$$\der {} x \left(\arctan w + \arctan \dfrac{1}{w}\right) = 0.$$

A function with zero derivative is constant, so

$$\arctan w + \arctan \dfrac{1}{w} = C, \quad\hbox{a constant}.$$

But when $w = 1$ ,

$$C = \arctan w + \arctan \dfrac{1}{w} = = \arctan 1 + \arctan 1 = \dfrac{\pi}{2}.$$

So I get the identity

$$\arctan w + \arctan \dfrac{1}{w} = \dfrac{\pi}{2}.\quad\halmos$$


Here are the integration formulas for some of the inverse trig functions. I'm giving extended versions of the formulas --- with "$a^2$ " replacing the "1" that you'd get if you just reversed the derivative formulas --- in order to save you a little time in doing problems.

$$\int \dfrac{1}{\sqrt{a^2 - x^2}}\,dx = \arcsin \dfrac{x}{a} + C$$

$$\int \dfrac{1}{a^2 + x^2}\,dx = \dfrac{1}{a} \arctan \dfrac{x}{a} + C$$

$$\int \dfrac{1}{|x|\sqrt{x^2 - a^2}}\,dx = \dfrac{1}{a} \arcsec \dfrac{x}{a} + C$$


Example. Derive the extended $\arcsin$ integral formula from the formula $\displaystyle \int \dfrac{1}{\sqrt{1 -
   x^2}}\,dx = \arcsin x + C$ .

$$\int \dfrac{1}{\sqrt{a^2 - x^2}}\,dx = \dfrac{1}{a} \int \dfrac{1} {\sqrt{1 - \left(\dfrac{x}{a}\right)^2}}\,dx = \dfrac{1}{a} \int \dfrac{1}{\sqrt{1 - u^2}}\cdot a\,du = \int \dfrac{du}{\sqrt{1 - u^2}} = \arcsin u + C = \arcsin \dfrac{x}{a} + C.$$

$$\left[u = \dfrac{x}{a}, \quad du = \dfrac{dx}{a}, \quad dx = a\,du\right]\quad\halmos$$


Example. Using the $\arctan$ formula with $a = 2$ ,

$$\int \dfrac{dx}{4 + x^2} = \dfrac{1}{2} \arctan \dfrac{x}{2} + C.$$

Using the $\arcsin$ formula with $a = \sqrt{3}$ ,

$$\int \dfrac{1}{\sqrt{3 - x^2}}\,dx = \sin^{-1} \dfrac{x}{\sqrt{3}} + C.\quad\halmos$$


Example.

$$\int \dfrac{dx}{1 + 4x^2} = \int \dfrac{dx}{1 + (2x)^2} = \int \dfrac{1}{1 + u^2}\cdot \dfrac{du}{2} = \dfrac{1}{2} \arctan u + C = \dfrac{1}{2} \arctan (2x) + C.$$

$$\left[u = 2x, \quad du = 2\,dx, \quad dx = \dfrac{du}{2}\right] \quad\halmos$$


Example.

$$\int \dfrac{x^4\,dx}{1 + x^{10}} = \int \dfrac{x^4\,dx}{1 + (x^5)^2} = \int \dfrac{x^4}{1 + u^2}\cdot \dfrac{du}{5x^4} = \dfrac{1}{5} \int \dfrac{du}{1 + u^2} =$$

$$\left[u = x^5, \quad du = 5x^4\,dx, \quad dx = \dfrac{du}{5x^4}\right]$$

$$\dfrac{1}{5} \arctan u + C = \dfrac{1}{5} \arctan (x^5) + C. \quad\halmos$$


Example.

$$\int \dfrac{e^x}{\sqrt{1 - e^{2x}}}\,dx = \int \dfrac{e^x}{\sqrt{1 - u^2}}\cdot \dfrac{du}{e^x} = \int \dfrac{du}{\sqrt{1 - u^2}} = \arcsin u + C = \arcsin e^x + C.$$

$$\left[u = e^x, \quad du = e^x\,dx, \quad dx = \dfrac{du}{e^x}\right] \quad\halmos$$


Example.

$$\int \dfrac{(\sec x)^2\,dx}{\sqrt{1 - (\tan x)^2}} = \int \dfrac{(\sec x)^2}{\sqrt{1 - u^2}}\cdot \dfrac{du}{(\sec x)^2} = \int \dfrac{du}{\sqrt{1 - u^2}} = \sin^{-1} u + C = \sin^{-1} \tan x + C.$$

$$\left[u = \tan x, \quad du = (\sec x)^2\,dx, \quad dx = \dfrac{du}{(\sec x)^2}\right]\quad\halmos$$


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