# Trigonometry : Complex Numbers/Polar Form

## Example Questions

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### Example Question #1 : De Moivre's Theorem And Finding Roots Of Complex Numbers

Simplify using De Moivre's Theorem:

Explanation:

We can use DeMoivre's formula which states:

Now plugging in our values of  and  we get the desired result.

### Example Question #2 : De Moivre's Theorem And Finding Roots Of Complex Numbers

Evaluate using De Moivre's Theorem:

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 #3 : De Moivre's Theorem And Finding Roots Of Complex Numbers

Use De Moivre's Theorem to evaluate .

Explanation:

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 #4 : De Moivre's Theorem And Finding Roots Of Complex Numbers

Use De Moivre's Theorem to evaluate .

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 #5 : De Moivre's Theorem And Finding Roots Of Complex Numbers

Find all fifth roots of .

Explanation:

Begin by converting the complex number to polar form:

Next, put this in its generalized form, using k which is any integer, including zero:

Using De Moivre's theorem, a fifth root of  is given by:

Assigning the values  will allow us to find the following roots. In general, use the values .

These are the fifth roots of .

### Example Question #6 : De Moivre's Theorem And Finding Roots Of Complex Numbers

Find all cube roots of 1.

Explanation:

Begin by converting the complex number to polar form:

Next, put this in its generalized form, using k which is any integer, including zero:

Using De Moivre's theorem, a fifth root of 1 is given by:

Assigning the values  will allow us to find the following roots. In general, use the values .

These are the cube roots of 1.

### Example Question #7 : De Moivre's Theorem And Finding Roots Of Complex Numbers

Find all fourth roots of .

Explanation:

Begin by converting the complex number to polar form:

Next, put this in its generalized form, using k which is any integer, including zero:

Using De Moivre's theorem, a fifth root of  is given by:

Assigning the values  will allow us to find the following roots. In general, use the values .

These are the fifth roots of .

### Example Question #1 : Complex Numbers/Polar Form

The polar coordinates  of a point are . Convert these polar coordinates to rectangular coordinates.

Explanation:

Given the polar coordinates , the  -coordinate is  . We can find this coordinate by substituting :

Likewise, given the polar coordinates , the  -coordinate is  .  We can find this coordinate by substituting :

Therefore the rectangular coordinates of the point  are .

### Example Question #2 : Complex Numbers/Polar Form

Express the complex number  in rectangular form.

Explanation:

To convert this number to rectangular form, first think about what  and  are equal to. Because , we can use a 30-60-90o reference triangle in the 3rd quadrant to determine these values.

Now plug these in and continue solving:

### Example Question #3 : Complex Numbers/Polar Form

For the complex number , find the modulus  and the angle . Then, express this number in polar form .