# Precalculus : Polar Coordinates and Complex Numbers

## Example Questions

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

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 #2 : 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:

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

Express the complex number  in polar form.

Explanation:

The figure below shows a complex number plotted on the complex plane. The horizontal axis is the real axis and the vertical axis is the imaginary axis.

The polar form of a complex number is . We want to find the real and complex components in terms of  and  where  is the length of the vector and  is the angle made with the real axis.

We use the Pythagorean Theorem to find :

We find  by solving the trigonometric ratio

Using ,

Then we plug  and  into our polar equation to obtain

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

What is the polar form of the complex number  ?

Explanation:

The polar form of a complex number  is  where  is the modulus of the complex number and  is the angle in radians between the real axis and the line that passes through  (   and ). We can solve for  and  easily for the complex number :

which gives us

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

Express the complex number in polar form:

Explanation:

Remember that the standard form of a complex number is: , which can be rewritten in polar form as: .

To find r, we must find the length of the line  by using the Pythagorean theorem:

To find , we can use the equation

Note that this value is in radians, NOT degrees.

Thus, the polar form of this equation can be written as