Stepwise problems - Sun. Dec 7, 11:59pm
Complex exponential
01
01
Complex forms - exponential to Cartesian
Write each number in the form
. (a)
(b) Link to originalSolution
01
(a) Use Euler’s Formula:
(b)
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02
02
Polar and exponential form
Write down Euler’s Formula.
Now write
: (i) in polar form
(ii) in exponential form Link to originalSolution
02
(The point is in Quadrant II, which is UNSAFE.)
So the polar form is:
The exponential form is:
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Complex roots
03
01
Complex roots using polar
Find the three cube (
) roots of . Write your answer in the form
. Link to originalSolution
03
For
roots, always start by writing in exponential: Now use the roots formula:
Write out these roots by evaluating
at : Link to original
Regular problems - Tue. Dec 9, 11:59pm
Complex exponential
04
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) Link to originalSolution
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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05
04
Complex products and quotients using polar
For each pair of complex numbers
and , compute: (a)
(b)
(Use polar forms with
.) Link to originalSolution
05
(a)
Product:
Dividend:
Reciprocal:
(b)
Product:
Dividend:
Reciprocal:
Link to original
06
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.)
Link to originalSolution
06
(a)
Convert to polar:
De Moivre’s Theorem:
(b)
Convert to polar:
De Moivre’s Theorem:
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Complex roots
07
02
Complex roots using polar
Find each of the indicated roots.
(a) The four
roots of . (b) The three cube (
) roots of . Try to write your answer in
form if that is not hard, otherwise leave it in polar form. Link to originalSolution
07
(a)
For
roots, always start by writing in exponential: Now use the roots formula:
Write out these roots by evaluating
at :
(b)
For
roots, always start by writing in exponential: Now use the roots formula:
Write out these roots by evaluating
at : (The polar answers are acceptable too.)
Link to original