# Precalculus : Polar Form of Complex Numbers

## Example Questions

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### Example Question #1 : Express Complex Numbers In Rectangular Form

Convert the following to rectangular form:

Explanation:

Distribute the coefficient 2, and evaluate each term:

### Example Question #1 : Express Complex Numbers In Rectangular Form

Convert the following to rectangular form:

Explanation:

Distribute the coefficient and simplify:

### Example Question #3 : Express Complex Numbers In Rectangular Form

Represent the polar equation:

in rectangular form.

Explanation:

Using the general form of a polar equation:

we find that the value of  is   and the value of  is .

The rectangular form of the equation appears as , and can be found by finding the trigonometric values of the cosine and sine equations.

distributing the 3, we obtain the final answer of:

### Example Question #4 : Express Complex Numbers In Rectangular Form

Represent the polar equation:

in rectangular form.

Explanation:

Using the general form of a polar equation:

we find that the value of  and the value of . The rectangular form of the equation appears as , and can be found by finding the trigonometric values of the cosine and sine equations.

Distributing the 4, we obtain the final answer of:

### Example Question #5 : Express Complex Numbers In Rectangular Form

Represent the polar equation:

in rectangular form.

Explanation:

Using the general form of a polar equation:

we find that the value of  and the value of . The rectangular form of the equation appears as , and can be found by finding the trigonometric values of the cosine and sine equations.

distributing the 5, we obtain the final answer of:

### Example Question #6 : Express Complex Numbers In Rectangular Form

Convert in rectangular form

Explanation:

To convert, just evaluate the trig ratios and then distribute the radius.

### Example Question #7 : Express Complex Numbers In Rectangular Form

Convert to rectangular form

Explanation:

To convert to rectangular form, just evaluate the trig functions and then distribute the radius:

### Example Question #8 : Express Complex Numbers In Rectangular Form

Convert to rectangular form

Explanation:

To convert, evaluate the trig ratios and then distribute the radius:

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

The following equation has complex roots:

Express these roots in polar form.

Explanation:

Every complex number can be written in the form a + bi

The polar form of a complex number takes the form r(cos  + isin

Now r can be found by applying the Pythagorean Theorem on a and b, or:

r =

can be found using the formula:

=

So for this particular problem, the two roots of the quadratic equation

are:

Hence, a = 3/2 and b = 3√3 / 2

Therefore r =  = 3

and  = tan^-1 (√3) = 60

And therefore x = r(cos  + isin )  = 3 (cos 60 + isin 60)

### Example Question #1 : Express Complex Numbers In Polar Form

Express the roots of the following equation in polar form.

Explanation:

First, we must use the quadratic formula to calculate the roots in rectangular form.

Remembering that the complex roots of the equation take on the form a+bi,

we can extract the a and b values.

We can now calculate r and theta.

Using these two relations, we get

. However, we need to adjust this theta to reflect the real location of the vector, which is in the 2nd quadrant (a is negative, b is positive); a represents the x-axis in the real-imaginary plane, b represents the y-axis.

The angle theta now becomes 150.

.

You can now plug in r and theta into the standard polar form for a number:

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