How Newton Computed Sines and Cosines Without a Calculator

Today, if we want to know the sine or cosine of an angle, we simply press a button on a calculator. But suppose there is no calculator, no computer, and no trigonometric table.

How could we actually compute, for example,

$$\sin (1°)$$

using only arithmetic?

One answer comes from the classical sine and cosine series associated with Isaac Newton. The series appeared in Newton’s De analysi per aequationes numero terminorum infinitas, written in 1665–1666. The elegant derivation below follows a later mean-value approach presented by Heinrich Dörrie, written here in modern calculus notation.

The goal

We want formulas that allow us to calculate the sine and cosine of an angle x. The angle must be measured in radians.

The formulas we will obtain are

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

and

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

Today these are familiar power series. What is especially interesting is that we can derive them from a very simple idea: repeatedly taking averages.

The key idea: average values

Consider the average value of the sine function on the interval from 0 to x. In modern calculus notation,

$$\frac{1}{x}{\int }_0^x\sin tdt=\frac{1-\cos x}{x}$$

Similarly, the average value of cosine is

$$\frac{1}{x}{\int }_0^x\cos tdt=\frac{\sin x}{x}$$

These two elementary formulas are enough to generate increasingly accurate approximations for both sine and cosine.

Start with the simplest inequality

For a positive angle x sufficiently close to zero,

$$\cos x<1$$

Now take the average of both sides from 0 to x. The average of the left side is

$$\frac{\sin x}{x}$$

while the average of 1 is simply 1. Therefore,

$$\frac{\sin x}{x}<1$$

and hence

$$\sin x<x$$

We have obtained our first approximation:

$$\sin x\approx x$$

Average again

Now start with

$$\sin x<x$$

and take averages once more. The average of the left side is

$$\frac{1-\cos x}{x}$$

while the average of the function t on the interval from 0 to x is x/2. Thus,

$$\frac{1-\cos x}{x}<\frac{x}{2}$$

Multiplying by x and rearranging gives

$$\cos x>1-\frac{x^2}{2!}$$

And again

Take averages of this new inequality. The average of cosine is

$$\frac{\sin x}{x}$$

and the average of the right-hand side is

$$1-\frac{x^2}{6}$$

Therefore,

$$\frac{\sin x}{x}>1-\frac{x^2}{6}$$

and consequently

$$\sin x>x-\frac{x^3}{3!}$$

We have now trapped sine between two simple expressions:

$$x-\frac{x^3}{3!}<\sin x<x$$

Keep repeating the process

Each time we take another average, we obtain another term. Continuing gives alternating upper and lower bounds:

$$\sin x<x-\frac{x^3}{3!}+\frac{x^5}{5!}$$

and then

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

The upper and lower approximations become closer and closer. Their common limiting value gives

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

The same procedure gives

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

A built-in error estimate

There is another important advantage. If we stop either series after a finite number of terms, the error is smaller in absolute value than the first term we leave out.

For example, if we use

$$\sin x\approx x-\frac{x^3}{3!}$$

then the error is smaller than

$$\frac{x^5}{5!}$$

This means that the series does not merely give us an approximation. It also tells us how accurate that approximation is.

Computing sin(1°)

Now let us actually calculate a trigonometric value without using a table.

One degree in radians is

$$x=\frac{\pi }{180}\approx 0.01745329252$$

Since this number is very small, just two terms of the sine series already give extraordinary accuracy:

$$\sin (1°)\approx x-\frac{x^3}{6}$$

Substituting the value of x gives

$$\sin (1°)\approx 0.0174524064$$

How accurate is this?

The first omitted term is

$$\frac{x^5}{120}<0.00000000002$$

Thus only two terms are enough to determine sin(1°) correctly to ten decimal places:

$$\sin (1°)\approx 0.0174524064$$

That is a remarkable amount of accuracy from such a short calculation.

Computing the cosine

The cosine works in exactly the same way. For one degree,

$$\cos x\approx 1-\frac{x^2}{2}+\frac{x^4}{24}$$

which gives

$$\cos (1°)\approx 0.9998476952$$

Again, only a few arithmetic operations are needed.

Why radians matter

There is one essential detail: these formulas require angles to be measured in radians.

The fundamental approximation

$$\sin x\approx x$$

works in this form because radian measure connects an angle directly with arc length on the unit circle.

If an angle is given in degrees, convert it first:

$$x=(\text{angle in degrees})\frac{\pi }{180}$$

The larger idea

What makes this argument beautiful is not merely the final formulas. It is how little we need in order to discover them.

We begin with the elementary inequality

$$\cos x<1$$

and repeatedly take average values. Each step produces another power of x, another factorial in the denominator, and a sharper approximation.

Eventually the familiar patterns emerge:

$$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$$ $$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots$$

A calculator evaluates sine and cosine instantly. These series reveal some of the mathematics behind that computation: trigonometric values can be constructed, digit by digit, using powers, factorials, addition, subtraction, multiplication, and division.

Reference

The mean-value derivation in this article is adapted, using modern calculus notation, from Heinrich Dörrie, 100 Great Problems of Elementary Mathematics: Their History and Solution, translated by David Antin, Dover Publications, Problem 15, “Newton’s Sine and Cosine Series,” pp. 59–63.

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Another Unexpected Side of Trigonometry

Newton’s approach shows that familiar trigonometric functions can be reconstructed from infinite algebraic expansions. But there is another surprising algebraic connection hiding in trigonometry.

Consider the finite sum $\cos \left(x\right)+\cos \left(2x\right)+\cdots +\cos \left(nx\right).$ At first glance, it has nothing to do with a geometric series. Yet a simple change of viewpoint reveals exactly that structure.

Continue exploring: A Sum of Cosines Is a Geometric Series—Could You Believe It?