How to Calculate Inverse Trig Functions Without a Calculator

Inverse trigonometric functions answer questions such as “which angle has sine $1/2$?” Their values can often be found exactly from the unit circle, even without a calculator. For less familiar inputs, identities and power series give ways to simplify or approximate the answer.

The key is to use the principal value: each inverse function returns an angle from a specific range. Without that restriction, the periodicity of sine, cosine, and tangent would make an inverse value ambiguous.

Know the principal ranges

The standard inverse functions are written $\arcsin x$, $\arccos x$, and $\arctan x$. Their domains and output ranges are:

Function	Input domain	Principal output range
$\arcsin x$	$-1\le x\le 1$	$-\frac{\pi}{2}\le y\le\frac{\pi}{2}$
$\arccos x$	$-1\le x\le 1$	$0\le y\le\pi$
$\arctan x$	Any real number	$-\frac{\pi}{2}<y<\frac{\pi}{2}$

For example, many angles have sine $1/2$, but $\arcsin(1/2)$ means the angle in the arcsine range. That angle is $\pi/6$, or $30^\circ$.

Also, $\sin^{-1}x$ is commonly used as another notation for $\arcsin x$; it does not mean the reciprocal $1/\sin x$.

Use unit-circle values

Memorize or derive the sine, cosine, and tangent values for the standard angles. For example, at $\pi/6$, $\pi/4$, and $\pi/3$, the sine values are $1/2$, $\sqrt{2}/2$, and $\sqrt{3}/2$, respectively. The cosine values occur in reverse order, and tangent is sine divided by cosine.

Then match the input to an angle and check that the angle lies in the inverse function’s principal range. A few examples:

$$\begin{aligned} \arcsin\left(\frac{\sqrt{3}}{2}\right)&=\frac{\pi}{3},\\ \arccos\left(-\frac{\sqrt{2}}{2}\right)&=\frac{3\pi}{4},\\ \arctan(\sqrt{3})&=\frac{\pi}{3}. \end{aligned}$$

For the second result, $3\pi/4$ is in the arccosine range $[0,\pi]$, and its cosine is $-\sqrt{2}/2$. The angle $-\pi/4$ has the same cosine, but it is outside that principal range, so it is not the value of $\arccos(-\sqrt{2}/2)$.

Simplify with identities

Several identities let you change the inverse function or the sign of the input:

$$\arccos x=\frac{\pi}{2}-\arcsin x,\qquad -1\le x\le 1.$$

Also, $\arcsin(-x)=-\arcsin x$ and $\arctan(-x)=-\arctan x$. These follow from the symmetry of sine and tangent. For cosine, which is even, use $\arccos(-x)=\pi-\arccos x$ on $[-1,1]$.

Worked example: Evaluate $\arccos(\sqrt{3}/2)$. Since $\arcsin(\sqrt{3}/2)=\pi/3$, the identity gives

$$\arccos\left(\frac{\sqrt{3}}{2}\right) =\frac{\pi}{2}-\frac{\pi}{3} =\frac{\pi}{6}.$$

This answer is in $[0,\pi]$, as required. If an expression involves a sum or multiple of angles, compound-angle identities can sometimes rewrite it in terms of standard values before you apply the inverse function.

Approximate unfamiliar inputs by hand

If the input is not a standard value, there may be no simple exact answer. A power series gives an approximation. For $|x|<1$, one useful expansion is

$$\arcsin x=x+\frac{x^3}{6}+\frac{3x^5}{40}+\frac{5x^7}{112}+\frac{35x^9}{1152}+\cdots.$$

Worked example: Approximate $\arcsin(0.2)$ in radians using the displayed terms:

$$\begin{aligned} \arcsin(0.2)&\approx 0.2+\frac{0.2^3}{6}+\frac{3(0.2)^5}{40} +\frac{5(0.2)^7}{112}+\frac{35(0.2)^9}{1152}\\ &\approx 0.2013579203. \end{aligned}$$

Thus $\arcsin(0.2)\approx 0.201358$ radians, or about $11.537^\circ$. Each term in this series is positive for this input, and the next term is about $0.00000000046$, so the displayed approximation is accurate to six decimal places.

The series is most useful when $|x|$ is comfortably below $1$. Near the endpoints, convergence is slower, so many terms may be needed. For arctangent, the expansion $\arctan x=x-x^3/3+x^5/5-x^7/7+\cdots$ works for $|x|\le 1$; for positive $x>1$, first use $\arctan x=\pi/2-\arctan(1/x)$.

Common mistakes

• Choosing an angle outside the principal range: Check the range after matching a trig value. This matters especially for arccosine.
• Confusing an inverse with a reciprocal: $\arcsin x$ and $1/\sin x$ are different operations.
• Using degrees and radians inconsistently: Exact answers such as $\pi/3$ are in radians. Convert with $180^\circ=\pi$ radians if you need degrees.
• Assuming every answer has a neat form: Unit-circle values often give exact answers, but a value such as $\arcsin(0.2)$ is usually reported as an approximation.