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Trigonometric Form of Complex Numbers

x plus y times i equals r times the sum of cosine theta and i times sine theta.

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Trigonometric Form of Complex Numbers

You have begun working with complex numbers in your math class. While describing numbers in the complex plane, you realize that the plotting of a complex number is a lot like plotting a set of points on a rectangular coordinate system.

You also learned in math class that you can convert coordinates from a rectangular system into a polar system. As you are considering this, you plot the complex number . Can you somehow convert this into a type of polar plot that you've done before?

Read on, and at the conclusion of this Concept, you'll be able to convert this number into a polar form.

Guidance

A number in the form , where and are real numbers, and is the imaginary unit, or , is called a complex number. Despite their names, complex numbers and imaginary numbers have very real and significant applications in both mathematics and in the real world. Complex numbers are useful in pure mathematics, providing a more consistent and flexible number system that helps solve algebra and calculus problems. We will see some of these applications in the examples throughout this Concept.

The following diagram will introduce you to the relationship between complex numbers and polar coordinates.

In the figure above, the point that represents the number was plotted and a vector was drawn from the origin to this point. As a result, an angle in standard position, , has been formed. In addition to this, the point that represents is units from the origin. Therefore, any point in the complex plane can be found if the angle and the value are known. The following equations relate and .

If we apply the first two equations to the point the result would be:

The right side of this equation is called the polar or trigonometric form of a complex number. A shortened version of this polar form is written as . The length is called the absolute value or the modulus, and the angle is called the argument of the complex number. Therefore, the following equations define the polar form of a complex number:

It is now time to implement these equations perform the operation of converting complex numbers in standard form to complex numbers in polar form. You will use the above equations to do this.

Example A

Represent the complex number graphically and express it in its polar form.

Solution: Here is the graph of .

Converting to polar from rectangular, and .

So, the polar form is .

Another widely used notation for the polar form of a complex number is . Finally, there is a third way to write a complex number, in the form of , where "r" is the length of the vector in polar form, and is the angle the vector makes with the positive "x" axis. This makes a total of three ways to write the polar form of a complex number.

Example B

Express the following polar form of each complex number using the shorthand representations.

a)

b)

Solution:

a)

b)

Example C

Represent the complex number graphically and give two notations of its polar form.

Solution: From the rectangular form of and

This is the reference angle so now we must determine the measure of the angle in the third quadrant.

One polar notation of the point is . Another polar notation of the point is

So far we have expressed all values of theta in degrees. Polar form of a complex number can also have theta expressed in radian measure. This would be beneficial when plotting the polar form of complex numbers in the polar plane.

The answer to the above example with theta expressed in radian measure would be:

Now that we have explored the polar form of complex numbers and the steps for performing these conversions, we will look at an example in circuit analysis that requires a complex number given in polar form to be expressed in standard form.

Guided Practice

1. The impedance , in ohms, in an alternating circuit is given by . Express the value for in standard form. (In electricity, negative angles are often used.)

2. Express the following complex numbers in their polar form.

3. Express the complex number graphically and write it in its polar form.

Solutions:

1. The value for is given in polar form. From this notation, we know that and Using these values, we can write:

Therefore the standard form is ohms.

2.

3.

Since is in the fourth quadrant then Expressed in polar form is or

Concept Problem Solution

You can now convert into polar form by using the equations giving the radius and angle of the number's position in the complex plane:

Therefore, the polar form of is .

Explore More

Plot each of the following points in the complex plane.

Find the trigonometric form of the complex numbers where .

Write each complex number in standard form.

Vocabulary Language: English

$rcis\theta$

$rcis\theta$

$rcis\theta$ is shorthand for the expression $r\cos\theta+ri\sin\theta$.
complex plane

complex plane

The complex plane is the graphical representation of the set of all complex numbers.
polar form

polar form

The polar form of a point or a curve is given in terms of $r$ and $\theta$ and is graphed on the polar plane.
rectangular form

rectangular form

The rectangular form of a point or a curve is given in terms of $x$ and $y$ and is graphed on the Cartesian plane.
trigonometric form

trigonometric form

To write a complex number in trigonometric form means to write it in the form $r\cos\theta+ri\sin\theta$. $rcis\theta$ is shorthand for this expression.
trigonometric polar form

trigonometric polar form

To write a complex number in trigonometric form means to write it in the form $r\cos\theta+ri\sin\theta$. $rcis\theta$ is shorthand for this expression.