# Further Applications of Trigonometry

## Polar Form of Complex Numbers

“God made the integers; all else is the work of man.” This rather famous quote by nineteenth-century German mathematician Leopold Kronecker sets the stage for this section on the polar form of a complex number. Complex numbers were invented by people and represent over a thousand years of continuous investigation and struggle by mathematicians such as Pythagoras, Descartes, De Moivre, Euler, Gauss, and others. Complex numbers answered questions that for centuries had puzzled the greatest minds in science.
We first encountered complex numbers in Complex Numbers. In this section, we will focus on the mechanics of working with complex numbers: translation of complex numbers from polar form to rectangular form and vice versa, interpretation of complex numbers in the scheme of applications, and application of De Moivre’s Theorem.

### Plotting Complex Numbers in the Complex Plane

Plotting a complex number 
  is similar to plotting a real number, except that the horizontal axis represents the real part of the number, 
  and the vertical axis represents the imaginary part of the number, 
 


### Finding the Absolute Value of a Complex Number

The first step toward working with a complex number in polar form is to find the absolute value. The absolute value of a complex number is the same as its magnitude, or 
  It measures the distance from the origin to a point in the plane. For example, the graph of 
  in , shows 
 


### Writing Complex Numbers in Polar Form

The polar form of a complex number expresses a number in terms of an angle 
  and its distance from the origin 
  Given a complex number in rectangular form expressed as 
  we use the same conversion formulas as we do to write the number in trigonometric form:
We review these relationships in .
We use the term modulus to represent the absolute value of a complex number, or the distance from the origin to the point 
  The modulus, then, is the same as 
  the radius in polar form. We use 
  to indicate the angle of direction (just as with polar coordinates). Substituting, we have

### Converting a Complex Number from Polar to Rectangular Form

Converting a complex number from polar form to rectangular form is a matter of evaluating what is given and using the distributive property. In other words, given 
  first evaluate the trigonometric functions 
  and 
  Then, multiply through by 
 


### Finding Products of Complex Numbers in Polar Form

Now that we can convert complex numbers to polar form we will learn how to perform operations on complex numbers in polar form. For the rest of this section, we will work with formulas developed by French mathematician Abraham De Moivre (1667-1754). These formulas have made working with products, quotients, powers, and roots of complex numbers much simpler than they appear. The rules are based on multiplying the moduli and adding the arguments.

### Finding Quotients of Complex Numbers in Polar Form

The quotient of two complex numbers in polar form is the quotient of the two moduli and the difference of the two arguments.

### Finding Powers of Complex Numbers in Polar Form

Finding powers of complex numbers is greatly simplified using De Moivre’s Theorem. It states that, for a positive integer 
  is found by raising the modulus to the 
  power and multiplying the argument by 
  It is the standard method used in modern mathematics.

### Finding Roots of Complex Numbers in Polar Form

To find the  in polar form, we use the 
  Root Theorem or De Moivre’s Theorem and raise the complex number to a power with a rational exponent. There are several ways to represent a formula for finding   roots of complex numbers in polar form.

### Key Concepts

1. Complex numbers in the form 
  are plotted in the complex plane similar to the way rectangular coordinates are plotted in the rectangular plane. Label the x-axis as the real axis and the y-axis as the imaginary axis. See .
2. The absolute value of a complex number is the same as its magnitude. It is the distance from the origin to the point: 
  See  and .
3. To write complex numbers in polar form, we use the formulas 
  and 
  Then, 
  See  and .
4. To convert from polar form to rectangular form, first evaluate the trigonometric functions. Then, multiply through by 
  See  and .
5. To find the product of two complex numbers, multiply the two moduli and add the two angles. Evaluate the trigonometric functions, and multiply using the distributive property. See .
6. To find the quotient of two complex numbers in polar form, find the quotient of the two moduli and the difference of the two angles. See .
7. To find the power of a complex number 
  raise 
   to the power 
   and multiply 
   by 
  See .
8. Finding the roots of a complex number is the same as raising a complex number to a power, but using a rational exponent. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, find the absolute value of the given complex number.
For the following exercises, write the complex number in polar form.
For the following exercises, convert the complex number from polar to rectangular form.
For the following exercises, find 
  in polar form.
For the following exercises, find 
  in polar form.
For the following exercises, find the powers of each complex number in polar form.
For the following exercises, evaluate each root.

### Graphical

For the following exercises, plot the complex number in the complex plane.

### Technology

For the following exercises, find all answers rounded to the nearest hundredth.