Further Complex Numbers — Topic Review

This review covers all four lessons: De Moivre’s Theorem Applications, Roots of Complex Numbers, Exponential Form and Euler’s Formula, and Loci in the Complex Plane.

Use De Moivre’s theorem to expand (cosθ + i sinθ)⁴ and hence express cos(4θ) in terms of cosθ only.
Compute (1 − i)¹⁰ using De Moivre’s theorem. Express your answer in the form a + bi.
Find all cube roots of 8i. Express each root in polar form and in exact Cartesian form.
Write z = 2e^i7π/6 in Cartesian form a + bi.
Use exponential form to simplify 3e^iπ/3 × 2e^iπ/4, and express the result in Cartesian form.
Solve z⁴ = −16. Find all four roots in exact Cartesian form.
Describe the locus |z − 2 + i| = |z − 4 − 3i| geometrically and find its Cartesian equation.
Use exponential form to compute (√3 + i)⁶. Express as a real number.
The five fifth roots of unity satisfy z⁵ = 1. Find all five roots, sketch their positions on the unit circle, and explain why their sum equals zero.
Sketch the region {z : 0 < arg(z − 1) ≤ π/2 and |z − 1| ≤ 3}.
Use Euler’s formula to show that cos(2θ) = (e^2iθ + e^−2iθ)/2, and hence evaluate ∫₀^π/4 cos(2θ) dθ.
Use De Moivre’s power reduction to express cos³θ in terms of cosθ and cos(3θ), then evaluate ∫₀^π/3 cos³θ dθ.
The locus of z satisfies |z − 2i|/|z − 2| = 1. Identify the locus and find its Cartesian equation.
The polynomial z⁶ − 64 = 0 has six roots. Find all six roots in exponential form, and verify that they form a regular hexagon on the Argand diagram.
Prove Euler’s identity e^iπ + 1 = 0 using Euler’s formula, and explain why all five fundamental constants (e, i, π, 1, 0) appear.

See Answers ➔

Further Complex Numbers — Topic Review

Use De Moivre’s theorem to expand (cosθ + i sinθ)4 and hence express cos(4θ) in terms of cosθ only.

Compute (1 − i)10 using De Moivre’s theorem. Express your answer in the form a + bi.

Find all cube roots of 8i. Express each root in polar form and in exact Cartesian form.

Write z = 2ei7π/6 in Cartesian form a + bi.

Use exponential form to simplify 3eiπ/3 × 2eiπ/4, and express the result in Cartesian form.

Solve z4 = −16. Find all four roots in exact Cartesian form.

Describe the locus |z − 2 + i| = |z − 4 − 3i| geometrically and find its Cartesian equation.

Use exponential form to compute (√3 + i)6. Express as a real number.

The five fifth roots of unity satisfy z5 = 1. Find all five roots, sketch their positions on the unit circle, and explain why their sum equals zero.

Sketch the region {z : 0 < arg(z − 1) ≤ π/2 and |z − 1| ≤ 3}.

Use Euler’s formula to show that cos(2θ) = (e2iθ + e−2iθ)/2, and hence evaluate ∫0π/4 cos(2θ) dθ.

Use De Moivre’s power reduction to express cos3θ in terms of cosθ and cos(3θ), then evaluate ∫0π/3 cos3θ dθ.

The locus of z satisfies |z − 2i|/|z − 2| = 1. Identify the locus and find its Cartesian equation.

The polynomial z6 − 64 = 0 has six roots. Find all six roots in exponential form, and verify that they form a regular hexagon on the Argand diagram.

Prove Euler’s identity eiπ + 1 = 0 using Euler’s formula, and explain why all five fundamental constants (e, i, π, 1, 0) appear.

Use De Moivre’s theorem to expand (cosθ + i sinθ)⁴ and hence express cos(4θ) in terms of cosθ only.

Compute (1 − i)¹⁰ using De Moivre’s theorem. Express your answer in the form a + bi.

Write z = 2e^i7π/6 in Cartesian form a + bi.

Use exponential form to simplify 3e^iπ/3 × 2e^iπ/4, and express the result in Cartesian form.

Solve z⁴ = −16. Find all four roots in exact Cartesian form.

Use exponential form to compute (√3 + i)⁶. Express as a real number.

The five fifth roots of unity satisfy z⁵ = 1. Find all five roots, sketch their positions on the unit circle, and explain why their sum equals zero.

Use Euler’s formula to show that cos(2θ) = (e^2iθ + e^−2iθ)/2, and hence evaluate ∫₀^π/4 cos(2θ) dθ.

Use De Moivre’s power reduction to express cos³θ in terms of cosθ and cos(3θ), then evaluate ∫₀^π/3 cos³θ dθ.

The polynomial z⁶ − 64 = 0 has six roots. Find all six roots in exponential form, and verify that they form a regular hexagon on the Argand diagram.

Prove Euler’s identity e^iπ + 1 = 0 using Euler’s formula, and explain why all five fundamental constants (e, i, π, 1, 0) appear.