## About "Multiplying complex numbers in exponential form"

Multiplying complex numbers in exponential form :

Here we are going to see how to multiply complex numbers in exponential form.

Case 1 :

(cos mθ + i sin mθ)  =   eimθ

Case 2 :

(cos mθ - i sin mθ)   =   e-imθ

By using one of the above methods, we may find the product of two or more complex numbers.

If the given complex number is in polar form, first we have to convert it into exponential form and multiply.

Let us look into some example problems based on the above concept.

Example 1 :

Find the product of following complex numbers

(cos 2θ + i sin 2θ) (cos 3θ + i sin 3θ)

Solution :

=  (cos 2θ + i sin 2θ) (cos 3θ + i sin 3θ)

By converting the above complex numbers written in the polar form into exponential form, we get

cos 2θ + i sin 2θ  =  ei2θ

cos 3θ + i sin 3θ  =  ei3θ

(cos 2θ + i sin 2θ) (cos 3θ + i sin 3θ)  =  ei2θ ⋅ ei3θ

=  ei(2θ + 3θ)

=  ei5θ

=  cos 5θ + i sin 5θ

Example 2 :

Find the product of following complex numbers

(cos 4θ + i sin 4θ)-6 (cos θ + i sin θ)8

Solution :

=  (cos 4θ + i sin 4θ)-6 (cos θ + i sin θ)8

By converting the above complex numbers written in the polar form into exponential form, we get

(cos 4θ + i sin 4θ)-6  =  (ei4θ)-6   =  e-i24θ

(cos θ + i sin θ)8 =  (eiθ)8  =  ei8θ

(cos 4θ + i sin 4θ)-6 (cos θ + i sin θ)=  e-i24θ ⋅ ei8θ

=  ei(-24 + 8)θ

=  ei(-16θ)

=  e-i 16θ

=  cos 16θ - i sin 16θ

Example 3 :

Find the product of following complex numbers

(cos 4θ + i sin 4θ)12 (cos 5θ - i sin 5θ)-6

Solution :

=  (cos 4θ + i sin 4θ)12 (cos 5θ - i sin 5θ)-6

=  (cos 4θ + i sin 4θ)12 (cos (-5θ) + i sin (-5θ))-6

=  (cos 12(4θ) + i sin 12(4θ)) (cos -6(-5θ) + i sin -6(-5θ))

=  (cos 48θ + i sin 48θ) (cos 30θ + i sin 30θ)

=  cos (48+30)θ +  i sin (48+30)θ

=  cos 78θ +  i sin 78θ

Example 4 :

Find the product of following complex numbers

(cos α + i sin α)3/(sin β + i cos β)4

Solution :

=  (cos α + i sin α)3/(sin β + i cos β)4

The denominator is not in the polar form. In order to simplify this, first we have to convert the denominator into polar form.

(sin β + i cos β)=  [cos (90 - β) + i sin (90 - β)]4

=  (cos α + i sin α)3/[cos (90 - β) + i sin (90 - β)]4

=  (cos 3α + i sin 3α)/cos 4(90 - β) + i sin 4(90 - β)

=  cos (3α - (360 - 4β)) + i sin (3α - (360 - 4β))

=  cos (3α - 360 + 4β) + i sin (3α - 360 + 4β)

=  cos ((3α + 4β) - 360) + i sin ((3α + 4β) - 360)

=  cos (360 - (3α + 4β)) + i sin (360 - (3α + 4β))

=  cos (3α + 4β) + i sin (3α + 4β)

After having gone through the stuff given above, we hope that the students would have understood "Product of two complex numbers in polar form".

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here. WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6