Polar Form of a Complex Number
In short: The polar form of a complex number swaps its two rectangular parts a + bi for a range r and a bearing θ, giving r(cos θ + i sin θ) as an equivalent name for the same point.
PrecalculusNumbers and OperationsN-CN.B.4
What is Polar Form of a Complex Number?
The polar form of a complex number describes a point by how far away it is and which way it lies, like giving a ship’s range and bearing rather than its east-west and north-south offsets.
For \(z = a + bi\), the modulus is \(r = \sqrt{a^2 + b^2}\) and the argument \(\theta\) satisfies \(\cos\theta = \frac{a}{r}\) and \(\sin\theta = \frac{b}{r}\). Then \(z = r(\cos\theta + i\sin\theta)\), often abbreviated \(r\,\text{cis}\,\theta\). The angle is usually chosen between \(0\) and \(2\pi\), and adding \(2\pi\) gives the same number.
How to spot it
- Products, quotients or powers are needed, and the polar form of a complex number makes them far easier.
- The task asks for the modulus and argument, or the distance and angle from the origin.
- A number is given as r(cos θ + i sin θ) and you must return to a + bi.
- The complex plane is drawn with a point and a ray from the origin.
The method
- Identify the real part \(a\) and the imaginary coefficient \(b\).
- Compute the modulus \(r = \sqrt{a^2 + b^2}\).
- Find the quadrant from the signs of \(a\) and \(b\), then use the reference angle \(\tan^{-1}\left|\frac{b}{a}\right|\) to get \(\theta\).
- Write the polar form of the complex number, \(r(\cos\theta + i\sin\theta)\), and test it by expanding.
Polar Form of a Complex Number: solved example
Problem
Write \(z = 3\sqrt{3} – 3i\) in polar form.
Solution
The modulus is \(r = \sqrt{(3\sqrt{3})^2 + (-3)^2} = \sqrt{27 + 9} = 6\). The point has a positive real part and a negative imaginary part, so it is in Quadrant IV.
The reference angle is \(\tan^{-1}\frac{3}{3\sqrt{3}} = 30^{\circ}\), so \(\theta = 330^{\circ}\). Therefore \(z = 6(\cos 330^{\circ} + i\sin 330^{\circ})\).
Try it
Question 1
Convert \(4(\cos 90^{\circ} + i\sin 90^{\circ})\) to the form \(a + bi\).
Show the answer
\(4(0 + i\cdot 1) = 4i\).
Question 2
State the modulus and argument for \(z = -5\).
Show the answer
The modulus is 5, and the point lies on the negative real axis, so the argument is \(\pi\).
Polar Form of a Complex Number vs. Polar Coordinates
Polar coordinates \((r, \theta)\) name a point in an ordinary plane. The form discussed here applies the same idea to a number in the complex plane, where multiplication also becomes meaningful: moduli multiply and arguments add. The picture is shared, but the second one carries arithmetic.
Keep going
Lessons
Worksheets and quizzes
- Polar Form and De Moivre’s Theorem – Worksheet
- Polar Form and De Moivre’s Theorem – Quiz
- Complex Numbers and Complex Zeros – Worksheet
- Complex Numbers and Complex Zeros – Quiz
Related words
- De Moivre's Theorem
- Polar Coordinates
- Complex Number
- Imaginary Unit (i)
- Complex Conjugate
- Unit Circle
Whole dictionary: A–Z · Precalculus · Numbers and Operations