01
Complex forms - exponential to Cartesian
Write each number in the form
. (a)
(b)
Solution
01
(a) Use Euler’s Formula:
(b)
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02
Polar and exponential form
Write down Euler’s Formula.
Now write
: (i) in polar form
(ii) in exponential form
Solution
02
(The point is in Quadrant II, which is UNSAFE.)
So the polar form is:
The exponential form is:
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03
Polar and exponential form
Write down Euler’s Formula.
Now write each of the following complex numbers (i) in polar form, and (ii) in exponential form.
(a)
(b)
Solution
04
(a)
(The point is in Quadrant IV, which is SAFE.)
So the polar form is:
The exponential form is:
(b)
This one is easy:
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04
Complex products and quotients using polar
For each pair of complex numbers
and , compute: (a)
(b)
(Use polar forms with
.)
Solution
05
(a)
Product:
Dividend:
Reciprocal:
(b)
Product:
Dividend:
Reciprocal:
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05
Complex powers using polar
Using De Moivre’s Theorem, write each number in the form
. (a)
(b) (First convert to polar/exponential, then compute the power, then convert back.)
Solution
06
(a)
Convert to polar:
De Moivre’s Theorem:
(b)
Convert to polar:
De Moivre’s Theorem:
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