Angles and polar form

Radians, sine and cosine on a right triangle and on the unit circle, and the polar form of a complex number in which multiplication adds angles.

August 31, 2026

9 minutes read

1853 words

Contents12

A point of the plane can be named by two coordinates, or by one distance and one turn. The second naming is the polar form, and it is the one under which multiplication of complex numbers becomes easy to read. Reaching it takes an angle measure and two ratios.

Assumed knowledge

Arithmetic and square roots. A right angle, and that a triangle has three sides.

Complex numbers appear from the fourth section onward. They are the subject of Numbers and sets , and a reader who wants only sine and cosine can stop after the third section.

Ground covered

Degrees and radians, and why the second measure is the one the formulas use. Sine and cosine as ratios of the sides of a right-angled triangle. The same two values as the coordinates of a point on a circle of radius one. The modulus of a complex number and its argument, which together are its polar form. Why multiplying two complex numbers multiplies their moduli and adds their arguments. The exponential notation that records that fact in one symbol. Last, an angle that advances at a fixed rate, which is where a sine wave comes from.

A reader who can already convert between $a + bi$ and $r e^{i\theta}$ can skip the page.

Degrees and radians

An angle measures a turn. The everyday measure divides a full turn into 360 equal degrees, so a quarter turn is 90 degrees.

The number 360 is a convention inherited from Babylonian arithmetic. It has no geometric meaning, so formulas written in degrees need conversion factors.

The second measure takes its unit from the circle. Draw a circle of radius one and mark an angle at its centre. The angle in radians is the length of the arc it cuts off. A whole turn cuts off the whole circumference, which is $2\pi$ , so a whole turn is $2\pi$ radians and a quarter turn is $\pi/2$ .

2\pi \text{ radians} = 360 \text{ degrees} \tag{1}

Every angle below is in radians unless a figure says degrees.

Sine and cosine

A right-angled triangle has one angle of a quarter turn. Choose one of its other two angles and call it $\theta$ . The side opposite $\theta$ is the opposite, the shorter side touching it is the adjacent, and the longest side, opposite the right angle, is the hypotenuse.

Two ratios of those sides are named.

\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \qquad \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \tag{2}

The 3-4-5 triangle. Every length can be checked by counting squares, and the two ratios come out as 0.8 and 0.6.

The ratios depend on the angle alone. Doubling every side of the triangle doubles the top and the bottom of each fraction and leaves both values unchanged. So $\cos\theta$ and $\sin\theta$ are properties of the angle, not of the particular triangle drawn.

The definition in equation (2) reaches only angles smaller than a quarter turn, because no larger angle fits inside a right-angled triangle. The next section removes that limit.

Unit circle

Setting the hypotenuse to 1 makes the two ratios into two lengths. The adjacent side becomes $\cos\theta$ and the opposite side becomes $\sin\theta$ , because dividing by 1 changes nothing.

Draw that triangle with its corner at the origin and its hypotenuse along a radius of the circle of radius one. The far end of the radius is the point with coordinates $(\cos\theta, \sin\theta)$ . That circle is the unit circle.

On a circle of radius one the two ratios are the coordinates of the point, and the angle is the length of the arc.

The radius can now be swung the whole way round, so every angle has a sine and a cosine. Past a quarter turn the cosine goes negative, because the point has moved to the left of the vertical axis. Past a half turn the sine goes negative as well.

Turning by a further whole turn returns the point to where it started, so both values repeat every $2\pi$ .

The two coordinates are the sides of a right-angled triangle whose hypotenuse is the radius. Pythagoras therefore holds between them, for every angle.

\cos^2\theta + \sin^2\theta = 1 \tag{3}

Modulus

A complex number $a + bi$ is drawn at the point with horizontal coordinate $a$ and vertical coordinate $b$ . Its distance from the origin is called its modulus, written $|z|$ and often $r$ .

The two coordinates are the sides of a right-angled triangle whose hypotenuse runs from the origin to the point, so Pythagoras gives the distance.

|a + bi| = \sqrt{a^2 + b^2} \tag{4}

The modulus of 3 + 4i is 5, because 9 and 16 make 25.

The modulus is never negative, and it is zero only for the number zero.

Polar form

The modulus says how far the point lies from the origin. It does not say in which direction. The direction is the angle from the positive horizontal axis to the line joining the origin to the point, measured anticlockwise. That angle is the argument of the number, written $\arg z$ .

A distance and a direction locate the point exactly, so $r$ and $\theta$ name it as completely as $a$ and $b$ do.

One point, named twice. Either the pair 3 and 2, or the distance 3.61 at an angle of 33.7 degrees.

Reading the coordinates off the distance and the angle is what the unit circle already gives. A point at distance $r$ is the point at distance 1 scaled by $r$ , so both its coordinates scale by $r$ .

a = r\cos\theta, \qquad b = r\sin\theta \tag{5}

Substituting equation (5) into $a + bi$ gives the polar form.

z = r(\cos\theta + i\sin\theta) \tag{6}

Going the other way, $r$ comes from equation (4) and $\theta$ from the ratio $b/a$ . The ratio alone is ambiguous, because a point and its opposite through the origin give the same ratio. Software resolves it with a two-argument arctangent, atan2(b, a), which reads the signs of both coordinates and returns an angle in the correct quarter of the plane.

The argument is fixed only up to whole turns, since adding $2\pi$ returns the same point. A single value is usually chosen from the range $-\pi$ to $\pi$ , which is what atan2 returns.

Multiplication as turning

Equation (3) of Numbers and sets multiplies two complex numbers by expanding brackets. In polar form the same operation reads as one instruction.

Two identities of trigonometry are needed. They give the sine and cosine of a sum of angles.

\begin{aligned} \cos(\alpha + \beta) &= \cos\alpha\cos\beta - \sin\alpha\sin\beta \\ \sin(\alpha + \beta) &= \sin\alpha\cos\beta + \cos\alpha\sin\beta \end{aligned} \tag{7}

Write the two numbers in polar form, multiply them out, and collect the real and imaginary parts.

\begin{aligned} z_1 z_2 &= r_1 r_2 (\cos\theta_1 + i\sin\theta_1)(\cos\theta_2 + i\sin\theta_2) \\ &= r_1 r_2 \left[ (\cos\theta_1\cos\theta_2 - \sin\theta_1\sin\theta_2) + i(\sin\theta_1\cos\theta_2 + \cos\theta_1\sin\theta_2) \right] \\ &= r_1 r_2 \left[ \cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2) \right] \end{aligned} \tag{8}

The last line is equation (6) again, with modulus $r_1 r_2$ and argument $\theta_1 + \theta_2$ . Multiplying two complex numbers multiplies their distances and adds their turns.

Multiplying by 2i turns the plane by a quarter and doubles every distance. The two arcs are drawn end to end, so the sum of the angles is visible.

The imaginary unit has modulus 1 and argument $\pi/2$ . Multiplying by $i$ therefore changes no distance and turns by a quarter. Doing it twice turns by a half. A half turn carries a point to its negative, which is equation (1) of Numbers and sets drawn rather than written.

Exponential form

Adding exponents is what multiplying powers already does, since $x^m x^n = x^{m+n}$ . Equation (8) adds angles when it multiplies. Writing the angle as an exponent therefore makes the two operations the same operation.

The notation that does so is Euler’s formula.

e^{i\theta} = \cos\theta + i\sin\theta \tag{9}

The right-hand side is the point at angle $\theta$ on the unit circle, from equation (6) with $r = 1$ . The formula states that the exponential function returns that point for an imaginary exponent. It is proved by comparing the power series of the three functions.¹

Equation (6) becomes shorter, and equation (8) becomes a statement about exponents.

z = r e^{i\theta}, \qquad z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)} \tag{10}

Repeated multiplication by the same number follows at once. Multiplying $n$ copies multiplies $n$ moduli and adds $n$ equal angles.

z^n = r^n e^{i n \theta} \tag{11}

So a complex number of modulus greater than 1 has powers that run outward and turn, and one of modulus less than 1 has powers that spiral in. A number on the unit circle has powers that only turn.

An angle that runs with time

Take an argument that grows by a fixed amount at every tick of a clock.

\theta_n = \theta_0 + n\,\Delta\theta \tag{12}

The point $e^{i\theta_n}$ then goes round the unit circle at a steady rate. Its real part, $\cos\theta_n$ , rises and falls between $1$ and $-1$ , which is a cosine wave read one sample at a time. With $f_s$ ticks per second, a full turn every $f_s / f$ ticks means $f$ turns per second, so the step is fixed by $f$ .

\Delta\theta = \frac{2\pi f}{f_s} \tag{13}

That $f$ is called the frequency of the wave, measured in hertz. The angle matters only up to whole turns, so a program keeps it between $0$ and $2\pi$ by subtracting $2\pi$ whenever it passes.

Two such angles running at rates $f_1$ and $f_2$ return together to where they both started only when $f_1 / f_2$ is a fraction of whole numbers. When the ratio is irrational the pair never repeats, and a process driven by both never quite revisits a state it has been in before. A designer who wants motion that does not loop picks rates in no simple ratio.

Signals and samples uses equations (12) and (13) as the definition of a sine wave.

Figure sources

All five drawings come from one script kept beside this page. One grid helper serves every figure, so the paper a reader learns in the first is the paper the last argues on. The first three figures use the 3-4-5 triangle, whose sides are whole numbers and can be checked by counting squares. The script needs NumPy and Matplotlib.

planes.py

uv run --with numpy --with matplotlib python3 planes.py