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The image describes the expression for a complex number \( z \) in different forms. The given equation is:

\[
z = a + ib = r \cos \phi + i r \sin \phi = r (\cos \phi + i \sin \phi)
\]

This equation represents the transition between the Cartesian form and the polar form of a complex number:

1. \( z = a + ib \) is the Cartesian form (rectangular form), where \( a \) is the real part and \( b \) is the imaginary part.
2. \( z = r (\cos \phi + i \sin \phi) \) is the polar or trigonometric form, where:
   - \( r \) is the modulus (or absolute value) of the complex number, \( r = \sqrt{a^2 + b^2} \).
   - \( \phi \) is the argument (or phase) of the complex number, which is the angle between the real axis and the line representing the complex number in the complex plane.

The expression \( r (\cos \phi + i \sin \phi) \) is known as the **polar form of a complex number**.

### Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:
1. What is the geometric interpretation of the polar form of a complex number?
2. How can you calculate the modulus \( r \) of a complex number?
3. What is the relationship between the Cartesian and polar forms of a complex number?
4. How do you compute the argument \( \phi \) of a complex number?
5. What is Euler's formula, and how does it relate to the polar form of a complex number?

**Tip:** When converting between Cartesian and polar forms, always ensure that the angle \( \phi \) is in the correct quadrant based on the signs of \( a \) and \( b \).

What is the polar form of a complex number?

Learn about the polar form of complex numbers, including how to express a complex number in the form z = r(cos(φ) + i*sin(φ)). Understand the key formulas and relationships between Cartesian and polar forms.

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