MULTIPLYING COMPLEX NUMBERS IN EXPONENTIAL FORM

Example 1 :

Find the product of the following complex numbers :

(cos2θ + isin2θ)(cos3θ + isin3θ)

Solution :

Convert the above complex numbers to exponential form. 

cos2θ + isin2θ = e^i2θ

cos3θ + isin3θ = e^i3θ

(cos2θ + isin2θ)(cos3θ + isin3θ) :

= e^i2θ ⋅ e^i3θ

= e^{i2θ + i3θ}

= e^{i(2θ + 3θ)}

= e^i5θ

= cos5θ + isin5θ

Example 2 :

Find the product of the following complex numbers :

(cos4θ + isin4θ)^-6(cosθ + isinθ)⁸

Solution :

Convert the above complex numbers to exponential form. 

cos4θ + isin4θ = e^i4θ

cosθ + isinθ = e^iθ

(cos4θ + isin4θ)^-6(cosθ + isinθ)⁸ :

= (e^i4θ)^-6 ⋅ (e^iθ)⁸

= e^-i24θ ⋅ e^i8θ

= e^{-i24θ + i8θ}

= e^{i(-24θ + 8θ)}

= e^i(-16θ)

  = e^-i16θ

  = cos16θ - isin16θ

Example 3 :

Find the product of the following complex numbers :

(cos4θ + isin4θ)¹²(cos5θ - isin5θ)^-6

Solution :

Convert the above complex numbers to exponential form. 

cos4θ + isin4θ = e^i4θ

cos5θ - isin5θ = e^-i5θ

(cos4θ + isin4θ)¹²(cos5θ - i sin5θ)^-6 :

= (e^i4θ)¹² ⋅ (e^-i5θ)^-6

= e^i48θ ⋅ e^i30θ

= e^{i48θ + i30θ}

= e^{i(48θ + 30θ)}

= e^i78θ

= cos78θ + isin78θ 

Example 4 :

Find the division of the following complex numbers 

(cosα + isinα)³/(sinβ + icosβ)⁴

Solution :

In the above division, complex number in the denominator is not in polar form.

First, convert the complex number in denominator to polar form. 

sinβ + icosβ = cos(90 - β) + isin(90 - β)

Then,

(cosα + isinα)³/(sinβ + icosβ)⁴ :

= (cosα + isinα)³/[cos(90 - β) + isin(90 - β)]⁴

Convert the above complex numbers to exponential form. 

cosα + isinα = e^i4α

cos(90° - β) + isin(90° - β) = e^{i(90° - β)}

Then,

= (e^i4α)³/(e^{i(90° - β)})⁴

= (e^i12α)/(e^{i(360° - 4β)})

Write the division as multiplication by changing the sign of the exponent in denominator. 

= (e^i12α) ⋅ (e^{-i(360° - 4β)})

= e^{i12α - i(360° - 4β)}

= e^{i[4α - (360° - 4β)]}

= e^{i[4α - 360° + 4β)]}

= e^{i[4α + 4β - 360°]}

= cos(4α + 4β - 360°) + isin(4α + 4β - 360°)

= cos[-(360-(4α + 4β))] + isin[-(360-(4α + 4β))]

= cos(4α + 4β) - isin[360-(4α + 4β)]

= cos(4α + 4β) + i sin(4α + 4β)

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