Trigonometry & Complex Numbers

Study Sheet

Trigonometry & Complex Numbers

Identities, de Moivre, and roots of unity

Trigonometric Identities

Tip
The identities you cannot do without

Pythagorean: sin⁡2θ+cos⁡2θ=1\sin^2\theta+\cos^2\theta=1. Double angle: sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta = 2\sin\theta\cos\theta, cos⁡2θ=1−2sin⁡2θ\cos 2\theta = 1 - 2\sin^2\theta. Law of cosines: c2=a2+b2−2abcos⁡Cc^2 = a^2+b^2-2ab\cos C.

In plain terms. A handful of identities convert between sines and cosines and relate a triangle's sides to its angles.

Example. If sin⁡θ=35\sin\theta = \tfrac35 (acute), then cos⁡θ=45\cos\theta = \tfrac45 and sin⁡2θ=2⋅35⋅45=2425\sin 2\theta = 2\cdot\tfrac35\cdot\tfrac45 = \tfrac{24}{25}.

Reminder — Law of Cosines:  c2=a2+b2−2abcos⁡Cc^{2}=a^{2}+b^{2}-2ab\cos C

Complex Numbers

Tip
Polar form and de Moivre

Write z=r(cos⁡θ+isin⁡θ)=reiθz = r(\cos\theta+i\sin\theta) = re^{i\theta}; then zn=rn(cos⁡nθ+isin⁡nθ)z^n = r^n(\cos n\theta + i\sin n\theta).

In plain terms. A complex number is a length and an angle. Raising to a power multiplies the length and just adds up the angle.

Example. (1+i)8(1+i)^8: here r=2, θ=45∘r=\sqrt2,\ \theta=45^\circ, so r8=16r^8 = 16 and 8θ=360∘8\theta = 360^\circ, giving 1616.

Concept
Roots of unity

The solutions of zn=1z^n=1 are e2πik/ne^{2\pi ik/n}, equally spaced on the unit circle, and for n>1n>1 they sum to 00.

In plain terms. The nnth roots of 11 are nn evenly-spaced points around a circle; being symmetric, they cancel to zero.

Example. The cube roots of 11 are 1,ω,ω21, \omega, \omega^2 with 1+ω+ω2=01 + \omega + \omega^2 = 0.