You already know how to represent complex numbers in the complex plane using rectangular coordinates and you already know how to multiply and divide complex numbers. Representing these points and performing these operations using trigonometric polar form will make your computations more efficient.
What are the two ways to multiply the following complex numbers?
Trigonometric Polar Form of Complex Numbers
Take the following complex number in rectangular form.
To convert the following complex number from rectangular form to trigonometric polar form, find the radius using the absolute value of the number.
The angle can be found with basic trig and the knowledge that the opposite side is always the imaginary component and the adjacent side is always the real component.
For basic problems, the amount of work required to compute products and quotients for complex numbers given in either form is roughly equivalent. For more challenging questions, trigonometric polar form becomes significantly advantageous.
In rectangular coordinates:
Convert the following complex number from trigonometric polar form to rectangular form.
Divide the following complex numbers.
Translate the following complex number from rectangular form into trigonometric polar form:
Note that this has no complex part and therefore has no angle.
Multiply the following complex numbers in trigonometric polar form.
Note how much easier it is to do products and quotients in trigonometric polar form.
Translate the following complex numbers from trigonometric polar form to rectangular form.
Translate the following complex numbers from rectangular form into trigonometric polar form.
Complete the following calculations and simplify.
To see the Review answers, open this PDF file and look for section 11.3.