Complex numbers as points: modulus, amplitude and trigonometric form
Answers are for parents and teachers.
1
Write the points of z = −2 + 3i, its conjugate, −z and 1 + z.
z = (−2, 3); conjugate −2 − 3i = (−2, −3); −z = 2 − 3i = (2, −3); 1 + z = −1 + 3i = (−1, 3).
2
Find the modulus and principal amplitude of (a) −6 + 2√3 i (b) 2 − 2i (c) −10i.
(a) |z| = √(36 + 12) = 4√3; α = tan⁻¹(2√3/6) = tan⁻¹(1/√3) = π/6; quadrant II: θ = 5π/6. (b) |z| = 2√2; α = π/4; quadrant IV: θ = −π/4. (c) |z| = 10, θ = −π/2.
3
Write −1 − √3 i and 12 in trigonometric form.
−1 − √3 i: r = 2, α = tan⁻¹(√3) = π/3, quadrant III, θ = π/3 − π = −2π/3: 2(cos(−2π/3) + i sin(−2π/3)). And 12 = 12(cos 0 + i sin 0).
4
Write 3(sin 50° + i cos 50°) in standard trigonometric form.
sin 50° = cos 40° and cos 50° = sin 40°, so it is 3(cos 40° + i sin 40°): modulus 3, amplitude 40°.
5
Write 8(cos 5π/6 + i sin 5π/6) as x + yi, and find z · z̄ for z = 5 − 12i.
8(−√3/2 + i/2) = −4√3 + 4i. And z · z̄ = (5 − 12i)(5 + 12i) = 25 + 144 = 169 = |z|².