Problem 97 - Algebra and Trigonometry

Question

If you are given a complex number in polar form, how do you write it in rectangular form?

Step-by-Step Solution

Answer

To convert a complex number from polar to rectangular form, apply the equations \( x = r \cos \theta \) and \( y = r \sin \theta \) to find the real part \( x \) and the imaginary part \( y \) respectively. Then, express the complex number in rectangular form as \( z = x + iy \).

Step 1: Identify the polar form of the complex number

The polar form of a complex number is typically given as \( r \angle \theta \). The complex number is represented by the magnitude \( r \) and the angle \( \theta \) in degrees or radians.

Step 2: Transform the polar form to rectangular form

To obtain the rectangular form \( z = x + iy \), apply the equations \( x = r \cos \theta \) and \( y = r \sin \theta \). The real part \( x \) is found by multiplying the magnitude \( r \) by the cosine of the angle \( \theta \). Similarly, the imaginary part \( y \) is found by multiplying the magnitude \( r \) by the sine of the angle \( \theta \). Note that \(\cos \theta \) points to the real part and \(\sin \theta \) points to the imaginary part.

Step 3: Write the rectangular form

After calculating \( x \) and \( y \), express the complex number in rectangular form as \( z = x + iy \). Note that \( i \) represents the imaginary unit.