Complex Numbers in Polar Form
Magnitude and argument.
Key idea
Magnitude and argument.
In depth
z = a + bi = r(cos θ + i sin θ), with r = |z| and θ the argument. Multiplication multiplies moduli and adds angles, which makes powers and roots easy.
Worked example
z = 1 + i = √2(cos 45° + i sin 45°). Then z² = 2(cos 90° + i sin 90°) = 2i.
Formula
Real-world uses
- Analysing alternating current with magnitude and phase in electrical engineering
Common mistake
Using the wrong quadrant for θ.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Complex Numbers in Polar Form?
Magnitude and argument. It is a topic at the expert level in Trigonometry.
What is the formula for Complex Numbers in Polar Form?
The key formula is z = r(cosθ + i sinθ).
Where is Complex Numbers in Polar Form used in real life?
Real-world examples: Analysing alternating current with magnitude and phase in electrical engineering.
What should I learn before Complex Numbers in Polar Form?
Make sure you are comfortable with Limaçons and Cardioids, the previous topic in Trigonometry.
What is a common mistake with Complex Numbers in Polar Form?
Using the wrong quadrant for θ.
How can I practise Complex Numbers in Polar Form?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Complex Numbers in Polar Form?
Move on to De Moivre’s Theorem next.
Related expert topics in Trigonometry
- Half Angle Formulas – Trig of θ/2.
- Power Reducing Formulas – Reduce squares to first powers of cos 2θ.
- Product-to-Sum Formulas – Convert products into sums.
- Sum-to-Product Formulas – Convert sums into products.
- Triple Angle Formulas – sin3θ and cos3θ in terms of θ.
- Trig and Inverse Trig Compositions – Evaluate expressions like sin(arccos x).