Trigonometry : De Moivre's Theorem and Finding Roots of Complex Numbers

Study concepts, example questions & explanations for Trigonometry

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Example Questions

Example Question #1 : Complex Numbers/Polar Form

Simplify using De Moivre's Theorem:

Possible Answers:

Correct answer:

Explanation:

We can use DeMoivre's formula which states:

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

Example Question #1 : Complex Numbers/Polar Form

Evaluate using De Moivre's Theorem: 

Possible Answers:

Correct answer:

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 #1 : Complex Numbers/Polar Form

Use De Moivre's Theorem to evaluate .

Possible Answers:

Correct answer:

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

Use De Moivre's Theorem to evaluate .

Possible Answers:

Correct answer:

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 #3 : Complex Numbers/Polar Form

Find all fifth roots of .

Possible Answers:

Correct answer:

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

Find all cube roots of 1.

Possible Answers:

Correct answer:

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

Find all fourth roots of .

Possible Answers:

Correct answer:

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 .

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