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Example Question #461 : Pre Calculus
Find the magnitude of the complex number described by .
To find the magnitude of a complex number we use the formula:
,
where our complex number is in the form .
Therefore,
Example Question #1 : Evaluate Powers Of Complex Numbers Using De Moivre's Theorem
Find the magnitude of :
, where the complex number satisfies
.
Note for any complex number z, we have:
.
Let . Hence
Therefore:
This gives the result.
Example Question #3 : Powers And Roots Of Complex Numbers
What is the magnitude of ?
To find the magnitude of a complex number we use the following formula:
, where
.
Therefore we get,
.
Now to find
.
Example Question #4 : Powers And Roots Of Complex Numbers
Simplify
We can use DeMoivre's formula which states:
Now plugging in our values of and
we get the desired result.
Example Question #5 : Powers And Roots Of Complex Numbers
First convert this point to polar form:
Since this number has a negative imaginary part and a positive real part, it is in quadrant IV, so the angle is
We are evaluating
Using DeMoivre's Theorem:
DeMoivre's Theorem is
We apply it to our situation to get.
which is coterminal with
since it is an odd multiplie
Example Question #6 : Powers And Roots Of Complex Numbers
Evaluate
First, convert this complex number to polar form:
Since the real part is positive and the imaginary part is negative, this is in quadrant IV, so the angle is
So we are evaluating
Using DeMoivre's Theorem:
DeMoivre's Theorem is
We apply it to our situation to get.
is coterminal with
since it is an even multiple of
Example Question #7 : Powers And Roots Of Complex Numbers
Evaluate
First convert the complex number into polar form:
Since the real part is negative but the imaginary part is positive, the angle should be in quadrant II, so it is
We are evaluating
Using DeMoivre's Theorem:
DeMoivre's Theorem is
We apply it to our situation to get.
simplify and take the exponent
is coterminal with
since it is an odd multiple of pi
Example Question #8 : Powers And Roots Of Complex Numbers
Use DeMoivre's Theorem to evaluate the expression .
First convert this complex number to polar form:
so
Since this number has positive real and imaginary parts, it is in quadrant I, so the angle is
So we are evaluating
Using DeMoivre's Theorem:
DeMoivre's Theorem is
We apply it to our situation to get.
Example Question #9 : Powers And Roots Of Complex Numbers
Evaluate:
First, convert this complex number to polar form.
Since the point has a positive real part and a negative imaginary part, it is located in quadrant IV, so the angle is .
This gives us
To evaluate, use DeMoivre's Theorem:
DeMoivre's Theorem is
We apply it to our situation to get.
simplifying
,
is coterminal with
since it is an even multiple of
Example Question #10 : Powers And Roots Of Complex Numbers
First, convert the complex number to polar form:
Since both the real and the imaginary parts are positive, the angle is in quadrant I, so it is
This means we're evaluating
Using DeMoivre's Theorem:
DeMoivre's Theorem is
We apply it to our situation to get.
First, evaluate . We can split this into
which is equivalent to
[We can re-write the middle exponent since is equivalent to
]
This comes to
Evaluating sine and cosine at is equivalent to evaluating them at
since
This means our expression can be written as:
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