How to Calculate Sine and Cosine Without a Calculator

You can calculate trigonometric values by hand in two ways: use exact identities for familiar angles, or find a numerical approximation for other angles. The most useful general method is to reduce the angle and then evaluate a Taylor series.

The first thing to check is whether the angle is given in degrees or radians. The standard Taylor series use radians; substituting a degree measure directly gives the wrong result.

Use exact values for familiar angles

The trigonometric functions have simple exact values at a few standard angles. Memorizing these makes it possible to calculate many other values using angle-addition formulas.

Degrees	Radians	$\sin x$	$\cos x$	$\tan x$
$0^\circ$	$0$	$0$	$1$	$0$
$30^\circ$	$\pi/6$	$1/2$	$\sqrt{3}/2$	$\sqrt{3}/3$
$45^\circ$	$\pi/4$	$\sqrt{2}/2$	$\sqrt{2}/2$	$1$
$60^\circ$	$\pi/3$	$\sqrt{3}/2$	$1/2$	$\sqrt{3}$
$90^\circ$	$\pi/2$	$1$	$0$	undefined

For example, the angle-addition identity

$$\sin(a+b)=\sin a\cos b+\cos a\sin b$$

lets you calculate $\sin 75^\circ$ exactly by writing $75^\circ=45^\circ+30^\circ$. Substituting the table values gives

$$\sin 75^\circ=\frac{\sqrt{2}}{2}\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}.$$

This approach is best when an angle can be built from familiar angles. It does not provide a short exact expression for every angle.

Reduce the angle before calculating

Trigonometric functions repeat and have symmetries. For sine and cosine,

$$\sin(x+2\pi k)=\sin x,\qquad \cos(x+2\pi k)=\cos x$$

for any integer $k$. Also, $\sin(-x)=-\sin x$ and $\cos(-x)=\cos x$. These identities let you move an angle into a smaller range before using a series.

If the angle is in degrees, convert it to radians using

$$x=\theta\frac{\pi}{180},$$

where $\theta$ is the degree measure. For instance, $30^\circ$ becomes $30\pi/180=\pi/6$ radians.

Use periodicity and quadrant symmetries to reduce the angle to a reference angle $r\in[0,\pi/2]$, while keeping track of the signs of sine and cosine. To ensure a Taylor-series argument no larger than $\pi/4$, if $r>\pi/4$, use cofunction identities: $\sin r=\cos(\pi/2-r)$ and $\cos r=\sin(\pi/2-r)$. The resulting series argument is then at most $\pi/4$.

Approximate with Taylor series

For an angle $x$ measured in radians, the series are

$$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots$$

and

$$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots.$$

Here, $n!$ means the product of the positive integers from $1$ through $n$. To approximate a value, calculate a few terms and stop when the next term is smaller than the accuracy you need. For a small angle, the terms decrease rapidly. In the alternating series, once terms are decreasing, the error from stopping is no larger than the first omitted term.

Worked example: approximate $\sin 1$

The input $1$ means one radian. Use the first five terms:

$$\begin{aligned} \sin 1 &\approx 1-\frac{1}{3!}+\frac{1}{5!}-\frac{1}{7!}+\frac{1}{9!}\\ &=1-\frac{1}{6}+\frac{1}{120}-\frac{1}{5040}+\frac{1}{362880}\\ &\approx 0.841471010. \end{aligned}$$

The next term is $1/11!\approx 0.0000000251$, so the error in this approximation is less than that amount. Thus $\sin 1\approx 0.84147$ to five decimal places.

For cosine, use its series in the same way. For example, the first four terms give $\cos 1\approx 1-1/2+1/24-1/720\approx 0.540278$. Add more terms if you need greater accuracy.

Calculate tangent and the other functions

Once you have sine and cosine, calculate tangent by division:

$$\tan x=\frac{\sin x}{\cos x},$$

provided $\cos x\ne 0$. For example, using the estimates above gives $\tan 1\approx 0.841468/0.540278\approx 1.557$.

The reciprocal functions follow directly:

$$\csc x=\frac{1}{\sin x},\qquad \sec x=\frac{1}{\cos x},\qquad \cot x=\frac{\cos x}{\sin x}.$$

These are undefined when their denominators are zero. You can also calculate other angles from a known value with identities such as $\sin(2x)=2\sin x\cos x$ and $\cos(2x)=\cos^2 x-\sin^2 x$.

Common mistakes

• Using degrees in a Taylor series. Convert to radians first. For example, use $\pi/6$, not $30$, for $30^\circ$.
• Ignoring the quadrant. Reducing an angle to a reference angle may change the sign. Sine and cosine signs depend on the original quadrant.
• Stopping too soon. The first term or two can give a rough estimate, but check the size of the next term to judge accuracy.
• Dividing by a value near zero. Tangent and reciprocal functions can be undefined or very sensitive to small rounding errors near their vertical asymptotes.

For a general angle, a decimal approximation is usually the practical result. Exact values are especially convenient for standard angles and angles related to them by identities.