What if you were given two complex numbers in polar form, such as
After completing this Concept, you'll know the Product Theorem, which will make it easier to multiply complex numbers.
Watch This
In the first part of this video you'll learn about the product of complex numbers in trigonometric form.
James Sousa: The Product and Quotient of Complex Numbers in Trigonometric Form
Guidance
Multiplication of complex numbers in polar form is similar to the multiplication of complex numbers in standard form. However, to determine a general rule for multiplication, the trigonometric functions will be simplified by applying the sum/difference identities for cosine and sine. To obtain a general rule for the multiplication of complex numbers in polar from, let the first number be
Now that the numbers have been designated, proceed with the multiplication of these binomials.
Therefore:
We can use this general formula for the product of complex numbers to perform computations.
Example A
Find the product of the complex numbers
Solution: Use the Product Theorem,
Example B
Find the product of
Solution: First, calculate
Example C
Find the product of the numbers
Solution:
First, convert
And now do the same with
And now substituting these values into the product theorem:
Guided Practice
1. Multiply together the following complex numbers. If they are not in polar form, change them before multiplying.
2. Multiply together the following complex numbers. If they are not in polar form, change them before multiplying.
3. Multiply together the following complex numbers. If they are not in polar form, change them before multiplying.
Solutions:
1.
2.
3.
Concept Problem Solution
Since you want to multiply
where
you can use the equation
and calculate:
This simplifies to:
Explore More
Multiply each pair of complex numbers. If they are not in trigonometric form, change them before multiplying.

3(cos32∘+isin32∘)⋅2(cos15∘+isin15∘) 
2(cos10∘+isin10∘)⋅10(cos12∘+isin12∘) 
4(cos45∘+isin45∘)⋅8(cos62∘+isin62∘) 
2(cos60∘+isin60∘)⋅12(cos34∘+isin34∘) 
5(cos25∘+isin25∘)⋅2(cos115∘+isin115∘) 
−3(cos70∘+isin70∘)⋅3(cos85∘+isin85∘) 
7(cos85∘+isin85∘)⋅2√(cos40∘+isin40∘) 
(3−2i)⋅(1+i) 
(1−i)⋅(1+i) 
(4−i)⋅(3+2i) 
(1+i)⋅(1+4i) 
(2+2i)⋅(3+i) 
(1−3i)⋅(2+i) 
(1−i)⋅(1−i)  Can you multiply a pair of complex numbers in standard form without converting to trigonometric form?