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If z1=r1(cos θ1+isin θ1) and z2=r2(cos θ2+isin θ2), then the product of these numbers is given as:
Notice that the product calls for multiplying the moduli and adding the angles.
Find the product of z1z2, given z1=4(cos(80°)+isin(80°)) and z2=2(cos(145°)+isin(145°)).
Follow the formula
The quotient of two complex numbers in polar form is the quotient of the two moduli and the difference of the two arguments.
If z1=r1(cos θ1+isin θ1) and z2=r2(cos θ2+isin θ2), then the quotient of these numbers is
Notice that the moduli are divided, and the angles are subtracted.
Given two complex numbers in polar form, find the quotient.
Find the quotient of z1=2(cos(213°)+isin(213°)) and z2=4(cos(33°)+isin(33°)).
Using the formula, we have
Find the product and the quotient of z1=2√3(cos(150°)+isin(150°)) and
Finding powers of complex numbers is greatly simplified using De Moivre’s Theorem . It states that, for a positive integer is found by raising the modulus to the power and multiplying the argument by It is the standard method used in modern mathematics.
If is a complex number, then
where is a positive integer.
Evaluate the expression using De Moivre’s Theorem.
Since De Moivre’s Theorem applies to complex numbers written in polar form, we must first write in polar form. Let us find
Then we find Using the formula gives
Use De Moivre’s Theorem to evaluate the expression.
To find the n th root of a complex number in polar form, we use the Root Theorem or De Moivre’s Theorem and raise the complex number to a power with a rational exponent. There are several ways to represent a formula for finding roots of complex numbers in polar form.
To find the root of a complex number in polar form, use the formula given as
where We add to in order to obtain the periodic roots.
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