# Precalculus : Polar Coordinates and Complex Numbers

## Example Questions

### Example Question #1 : Evaluate Powers Of Complex Numbers Using De Moivre's Theorem

Evaluate

Explanation:

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 .

Explanation:

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 #1 : Powers And Roots Of Complex Numbers

Evaluate:

Explanation:

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

Explanation:

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:

### Example Question #1 : Find The Roots Of Complex Numbers

Evaluate , where  is a natural number and  is the complex number .

Explanation:

Note that,

### Example Question #2 : Find The Roots Of Complex Numbers

What is the  length of

?

Explanation:

We have

.

Hence,

.

### Example Question #11 : Powers And Roots Of Complex Numbers

Solve for  (there may be more than one solution).

Explanation:

Solving that equation is equivalent to solving the roots of the polynomial .

Clearly, one of roots is 1.

Thus, we can factor the polynomial as

so that we solve for the roots of .

Using the quadratic equation, we solve for roots, which are .

This means the solutions to  are

### Example Question #1 : Find The Roots Of Complex Numbers

Recall that  is just shorthand for  when dealing with complex numbers in polar form.

### Express   in polar form.

Explanation:

First we recognize that we are trying to solve  where .

Then we want to convert  into polar form using,

and .

Then since De Moivre's theorem states,

if  is an integer, we can say

.

### Example Question #5 : Find The Roots Of Complex Numbers

Solve for  (there may be more than one solution).

Explanation:

To solve for the roots, just set equal to zero and solve for z using the quadratic formula () :  and now setting both  and  equal to zero we end up with the answers  and

Compute