Modulus-Argument Form (Edexcel A-Level Further Mathematics): Revision Notes

1.1.4 Modulus-Argument Form

What is Modulus-Argument Form?

A complex number $z = a + bi$ can be written in a modulus-argument form as:

$$z = r(\cos \theta + i \sin \theta)$$

where:

• $r = |z|$ is the modulus of the complex number (the distance from the origin to the point),
• $\theta = \arg(z)$ is the argument of the complex number (the angle with the positive real axis),
• $\cos \theta \ \text{and} \sin \theta$ represent the directions of the real and imaginary parts. This form is sometimes written as:

$$z = r \text{cis} \theta$$

where cis stands for $\cos \theta + i \sin \theta$

Converting to Modulus-Argument Form

To convert a complex number from Cartesian form $z = a + bi$ to modulus-argument form, follow these steps:

Method

Step 1: Find the modulus

Step 2: Find the argument

Step 3: Write the number in the form

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Step 1: Find the modulus using the formula:

$$r = |z| = \sqrt{a^2 + b^2}$$

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Step 2: Find the argument using:

$$\theta = \tan^{-1}\left(\frac{b}{a}\right)$$

Make sure you adjust the angle based on which quadrant the complex number lies in.

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Step 3: Write the number in the form:

$$z = r(\cos \theta + i \sin \theta)$$

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Note Summary