Polar Coordinates and Complex Numbers
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Pre-Calculus › Polar Coordinates and Complex Numbers
Convert the polar coordinates to rectangular form:
Explanation
To convert polar coordinates to rectangular coordinates
,
Using the information given in the question,
The rectangular coordinates are
Evaluate:
Explanation
To evaluate this problem we need to FOIL the binomials.
Now recall that
Thus,
Convert to polar form.
Explanation
Write the Cartesian to polar conversion formulas.
Substitute the coordinate point to the equations to find .
Since is not located in between the first quadrant, this is not the correct angle. The correct location of this coordinate is in the third quadrant. Add
radians to get the correct angle.
Therefore, the answer is .
What is the magnitude of ?
Explanation
To find the magnitude of a complex number we use the following formula:
, where
.
Therefore we get,
.
Now to find
.
Find the magnitude of :
, where the complex number satisfies
.
Explanation
Note for any complex number z, we have:
.
Let . Hence
Therefore:
This gives the result.
Write the equation in polar form
Explanation
First re-arrange the original equation so that the 4 is factored out on the right side, and put and
next to each other:
Make the substitutions and
:
take the square root of both sides
divide both sides by r
add
to both sides
Convert from polar form to rectangular form:
Explanation
Start by multiplying both sides by .
Keep in mind that
Remember that
So then,
Now, complete the square.
What is the magnitude of ?
Explanation
To find the magnitude of a complex number we use the following formula:
, where
.
Therefore we get,
.
Now to find
.
Evaluate:
Explanation
To evaluate this problem we need to FOIL the binomials.
Now recall that
Thus,
Evaluate:
Explanation
To evaluate this problem we need to FOIL the binomials.
Now recall that
Thus,