Complex numbers as points: modulus, amplitude and trigonometric form

Since x = r cos θ and y = r sin θ, the number z = x + yi equals r cos θ + i r sin θ = r(cos θ + i sin θ). This is the trigonometric (polar) form (in Arabic الصورة المثلثية). To write a number in this form: (1) find r = |z|; (2) find the reference angle and the quadrant; (3) write θ, the principal amplitude; (4) put r, cos θ and sin θ together, and check one part by multiplying back. For the standard form we need r positive, the same angle in cos and sin, and a plus sign between them. Expressions that only look like it must be rewritten: r(cos θ − i sin θ) = r(cos(−θ) + i sin(−θ)); a minus in front, −r(cos θ + i sin θ), adds or subtracts π to the angle; and sin θ + i cos θ swaps sine and cosine, which means turning to the angle 90° − θ (because sin θ = cos(90° − θ) and cos θ = sin(90° − θ)). The table repeats the exact values you need. Numbers on the axes are easy: 6 = 6(cos 0 + i sin 0), −6 = 6(cos π + i sin π), 4i = 4(cos π/2 + i sin π/2) and −2i = 2(cos(−π/2) + i sin(−π/2)).