How to Calculate Arcsine Without a Calculator

The arcsine, written $\arcsin(x)$ or $\sin^{-1}(x)$, asks: “Which angle has sine $x$?” You can often answer exactly by recognizing a familiar sine value. For other inputs, a power series gives a systematic approximation using arithmetic alone.

For real numbers, $x$ must be in $[-1,1]$. The principal value of arcsine is always between $-\pi/2$ and $\pi/2$, or between $-90^\circ$ and $90^\circ$. This range matters: although $\sin(150^\circ)=1/2$, the principal answer to $\arcsin(1/2)$ is $30^\circ$, not $150^\circ$.

Use familiar sine values for exact answers

Start by checking whether the input is a standard sine value. The most useful values are:

$x$	$\arcsin(x)$ in radians	$\arcsin(x)$ in degrees
$0$	$0$	$0^\circ$
$1/2$	$\pi/6$	$30^\circ$
$\sqrt{2}/2$	$\pi/4$	$45^\circ$
$\sqrt{3}/2$	$\pi/3$	$60^\circ$
$1$	$\pi/2$	$90^\circ$

Use the fact that arcsine is an odd function: $\arcsin(-x)=-\arcsin(x)$. So, for instance, $\arcsin(-\sqrt{3}/2)=-\pi/3=-60^\circ$.

Example: Since $\sin(30^\circ)=1/2$ and $30^\circ$ is in the principal range, $\arcsin(1/2)=30^\circ=\pi/6$. The radian and degree answers describe the same angle; the power series below gives answers in radians.

Approximate other inputs with a power series

For $|x|<1$, arcsine has the expansion

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

To estimate $\arcsin(x)$ by hand, calculate successive terms and add them. The terms get smaller for $|x|<1$. You can generate each term from the previous one without separately calculating its coefficient:

$$t_0=x,\qquad t_{n+1}=t_n x^2\frac{(2n+1)^2}{2(n+1)(2n+3)}.$$

Then add $t_0+t_1+t_2+\cdots$. For a negative input, the final answer is the negative of the answer for its positive counterpart.

Worked example: estimate $\arcsin(0.3)$. The first terms are

$$\begin{aligned} \arcsin(0.3)&\approx 0.3+\frac{0.3^3}{6}+\frac{3(0.3)^5}{40}+\frac{5(0.3)^7}{112}+\frac{35(0.3)^9}{1152}\\ &\approx 0.300000000+0.004500000+0.000182250+0.000009763+0.000000598\\ &\approx 0.304692611\text{ radians}. \end{aligned}$$

The next term is about $0.0000000396$. The ratio of each term’s magnitude to the preceding one is at most $x^2$, so after this next term the remaining tail is less than $0.0000000396/(1-0.3^2)$, or about $0.000000044$. Thus $0.30469265$ radians is a good approximation. To convert radians to degrees, multiply by $180/\pi$; this is about $17.46^\circ$.

Improve the series near an endpoint

The series converges more slowly as $|x|$ approaches $1$. At $x=1$ or $x=-1$, it still gives the correct limiting value, but adding terms by hand is inefficient. For a positive input near $1$, use

$$\arcsin(x)=\frac{\pi}{2}-\arcsin\!\left(\sqrt{1-x^2}\right),\qquad 0\le x\le 1.$$

This changes the input to a smaller number, where the series converges faster. For negative inputs, use odd symmetry. The identity is useful only if you can calculate the square root well enough for your purpose.

Common mistakes

• Ignoring the principal range: $\arcsin(x)$ returns only an angle from $-90^\circ$ to $90^\circ$. Other angles with the same sine are not its principal value.
• Mixing degrees and radians: The power series produces radians. Convert only after finding the approximate radian value.
• Using an input outside the domain: If $|x|>1$, there is no real angle whose sine is $x$, so the real-valued arcsine is undefined.
• Expecting a short series to work near $1$: Terms shrink slowly there. Use the endpoint identity when applicable, or include more terms and allow for the larger error.