Trigonometry and Complex Numbers

Chem 3240 · Appendix A.2

Davit Potoyan

One theme: rotation in a plane

• One point, two ways to name it: across and up, or out and around
• Trigonometry converts between them; complex numbers make the second one algebra

Sine and cosine are coordinates

• Rotate a point on the unit circle by \theta radians: x = \cos\theta, y = \sin\theta
• A full turn is 2\pi; each function is the other shifted by a quarter turn

Identities you will reach for

\cos^2\theta + \sin^2\theta = 1

Angle sum, the workhorse

\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B

\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B

• Do not memorize the table: every line falls out of Euler’s formula

Two more you will use constantly

Power reduction, a square becomes a single cosine

\sin^2\theta = \tfrac{1}{2}(1 - \cos 2\theta), \qquad \cos^2\theta = \tfrac{1}{2}(1 + \cos 2\theta)

Small angles in radians, the first Taylor terms

\sin\theta \approx \theta, \qquad \cos\theta \approx 1 - \tfrac{1}{2}\theta^2

• The second one is how a vibrating bond becomes a harmonic oscillator

Complex numbers live in 2D

• x^2 + 1 = 0 has no real root, so extend the line to a plane with one new symbol, i^2 = -1

z = x + iy, \qquad x = \operatorname{Re} z,\ \ y = \operatorname{Im} z

• A complex number is a point in a plane: real part across, imaginary part up
• 3 + 2i, -2i, and 1.1 are all complex numbers

Why we needed i: every root, every time

• Lift the parabola and the crossings vanish, but the roots do not: they step off the line
• Fundamental theorem of algebra: degree n means exactly n complex roots, always

Polar form and Euler’s formula

Locate z by its distance r and angle \phi

x = r\cos\phi, \qquad y = r\sin\phi

e^{i\phi} = \cos\phi + i\sin\phi

z = r(\cos\phi + i\sin\phi) = r\,e^{i\phi}

• r = \sqrt{x^2 + y^2} is the magnitude, \phi the phase

Example: Cartesian to polar

Write z = -4 + 4i in polar form.

r = \sqrt{(-4)^2 + 4^2} = 4\sqrt{2}, \qquad \phi = \frac{3\pi}{4}

z = 4\sqrt{2}\;e^{i\,3\pi/4}

• Second quadrant, so take \phi from arctan2(y, x), not from \arctan(y/x)
• On a calculator atan sees -4/4 and 4/{-4} as one ratio and always answers in quadrant I or IV; atan2 reads both signs

Multiplication is rotation

(r_1 e^{i\phi_1})(r_2 e^{i\phi_2}) = r_1 r_2\, e^{i(\phi_1 + \phi_2)}

• Magnitudes multiply, angles add
• |e^{i\phi}| = 1, so multiplying by e^{i\phi} is a pure turn by \phi: no stretching
• i = e^{i\pi/2} is a quarter turn; four of them bring you home, i^4 = 1
• e^{-i\phi} undoes e^{i\phi}; angles matter only mod 2\pi

Example: multiply the same idea two ways

Find (3 + 4i)(1 - 2i), then \left(2e^{i\pi/6}\right)\left(3e^{i\pi/3}\right).

(3 + 4i)(1 - 2i) = 3 - 6i + 4i - 8i^2 = 11 - 2i

\left(2e^{i\pi/6}\right)\left(3e^{i\pi/3}\right) = 6\,e^{i(\pi/6 + \pi/3)} = 6\,e^{i\pi/2} = 6i

• Cartesian: expand, then i^2 = -1. Polar: lengths multiply, angles add, and you are done

Live: dial the phase

z and e^{i\phi} z: the length never changes, only the angle

viewof phi = Inputs.range([-6.28, 6.28], {step: 0.05, value: 0.9, label: "rotation angle φ (rad)"})

{
  const z = {re: 1.3, im: 0.5};
  const w = {re: Math.cos(phi) * z.re - Math.sin(phi) * z.im, im: Math.sin(phi) * z.re + Math.cos(phi) * z.im};
  const R = Math.hypot(z.re, z.im);
  const circle = d3.range(0, 6.3, 0.05).map(t => ({x: R * Math.cos(t), y: R * Math.sin(t)}));
  const a0 = Math.atan2(z.im, z.re);
  const arc = d3.range(0, 1.0001, 0.02).map(s => ({x: 0.5 * Math.cos(a0 + s * phi), y: 0.5 * Math.sin(a0 + s * phi)}));
  return Plot.plot({
    width: 440, height: 440, aspectRatio: 1,
    x: {domain: [-2, 2], label: "Real", ticks: 5}, y: {domain: [-2, 2], label: "Imaginary", ticks: 5},
    marks: [
      Plot.line(circle, {x: "x", y: "y", stroke: "#bbb", strokeDasharray: "4,4"}),
      Plot.ruleX([0], {stroke: "#333"}), Plot.ruleY([0], {stroke: "#333"}),
      Plot.arrow([{x1: 0, y1: 0, x2: z.re, y2: z.im}], {x1: "x1", y1: "y1", x2: "x2", y2: "y2", stroke: "#e67e22", strokeWidth: 3}),
      Plot.arrow([{x1: 0, y1: 0, x2: w.re, y2: w.im}], {x1: "x1", y1: "y1", x2: "x2", y2: "y2", stroke: "#107895", strokeWidth: 3}),
      Plot.line(arc, {x: "x", y: "y", stroke: "#C8102E", strokeWidth: 2}),
      Plot.text([{x: z.re + 0.15, y: z.im + 0.1, t: "z"}], {x: "x", y: "y", text: "t", fill: "#e67e22", fontSize: 18}),
      Plot.text([{x: w.re + 0.15, y: w.im + 0.1, t: "e^(iφ) z"}], {x: "x", y: "y", text: "t", fill: "#107895", fontSize: 18})
    ]
  });
}

md`|z| = **${Math.hypot(1.3, 0.5).toFixed(2)}** before and after: a pure ${phi >= 0 ? "counterclockwise" : "clockwise"} turn by ${phi.toFixed(2)} rad`

The rotating phase is a helix

e^{i\omega t} = \cos\omega t + i\sin\omega t

• The tip runs around a circle in the complex plane; let time run along a third axis and the path is a helix
• Its shadow on one wall is \cos\omega t, on the other \sin\omega t
• One turning arrow carries both waves, which is why one exponential replaces two trig functions

The conjugate and the modulus

\bar{z} = x - iy = r\,e^{-i\phi}, \qquad |z|^2 = \bar{z}\,z = x^2 + y^2 = r^2

• Conjugation reflects z across the real axis; \bar z z rotates it back onto the axis and leaves a non-negative number
• Sums and differences of e^{\pm i\phi} hand the trig functions back:

\cos\phi = \frac{e^{i\phi} + e^{-i\phi}}{2}, \qquad \sin\phi = \frac{e^{i\phi} - e^{-i\phi}}{2i}

Example: modulus and division

Find |7 + 24i|, then divide 3 + 4i by 1 - 2i.

|7 + 24i| = \sqrt{49 + 576} = \sqrt{625} = 25

\frac{3 + 4i}{1 - 2i} = \frac{(3 + 4i)(1 + 2i)}{(1 - 2i)(1 + 2i)} = \frac{-5 + 10i}{5} = -1 + 2i

• To divide, multiply top and bottom by the conjugate of the bottom: the denominator turns real

Adding waves is adding arrows

Each cosine is the shadow of a turning arrow, \cos\theta = \operatorname{Re}\,e^{i\theta}

\cos\theta + \cos(\theta + \phi) = \operatorname{Re}\,e^{i\theta} + \operatorname{Re}\,e^{i\theta}e^{i\phi} = \operatorname{Re}\big[e^{i\theta} + e^{i\theta}e^{i\phi}\big]

\cos\theta + \cos(\theta + \phi) = \operatorname{Re}\big[e^{i\theta}\,(1 + e^{i\phi})\big]

• Two rules only: e^{i(\theta + \phi)} = e^{i\theta}e^{i\phi}, and \operatorname{Re} of a sum is the sum of the \operatorname{Re}s
• Factoring the shared e^{i\theta} leaves 1 + e^{i\phi}: two arrows a fixed angle \phi apart

Interference is the length of one arrow

|1 + e^{i\phi}| = 2\left|\cos(\phi/2)\right|

• \phi = 0: arrows aligned, amplitude doubles. \phi = \pi: opposed, amplitude zero
• Bright and dark fringes are exactly this arrow, long or short

Derivatives become multiplications

\frac{\partial}{\partial x}\,e^{i(kx - \omega t)} = ik\,e^{i(kx - \omega t)}, \qquad \frac{\partial}{\partial t}\,e^{i(kx - \omega t)} = -i\omega\,e^{i(kx - \omega t)}

• A wave is the real part of a rotating phase: \cos(kx - \omega t) = \operatorname{Re}\,e^{i(kx - \omega t)}
• Differentiating only brings down a constant factor; twice gives -k^2 and -\omega^2
• Taking the real part commutes with adding and differentiating, so do all the algebra on the exponential and take \operatorname{Re} at the very end

Takeaway

e^{i\phi} = \cos\phi + i\sin\phi turns trigonometry into exponentials: multiplying by e^{i\phi} rotates without stretching, \bar z z = |z|^2 turns any complex number into a real length, and adding waves becomes adding arrows.