Trigonometry Notes: Trigonometric Form of a Complex Numer

This material appears in section 6.5. Also, see Section 2.4 of the text for an introduction to Complex numbers.

• Rectangular Form
  A complex number is written in rectangular form

  

  where and are real numbers and is the imaginary unit.

  The "real part" of is the , and the "imaginary part" is the .

  Using the rule we can extend all the usual rules of arithmetic to complex numbers.

  • addition
    

    

  • subtraction
    

    

  • multiplication
    Remember that .

    

  • division
    There's a trick to division in rectangular coordinates

    

    The equation is true because we are just multiplying by , but this trick will make the denominator real:

    

• Trigonometric Form
  We can think of the Complex numbers as a plane where the real part gives the horizontal coordinate and the imaginary part gives the vertical coordinate. Points in the plane can be described also by distance from the origin and direction. Writing a complex number in trigonometric form is a way of expressing it in terms of magnitude and direction:

  

  where is the magnitude, which is a positive real number or 0, and is the angle. Notice the similarity to wiritng a vector as a magniutde times a unit vector written in trig form

  

  The idea here is the same and the computations are the same as well.

  The magnitude of a complex number (also called the norm or absolute value of is

  

  To compute the direction (also called the argument of ) we do the same calculation as we did in finding the direction of a vector:

  

• Converting from one form to the other
  
  • Trig to Rectangular
    This one is easy, just distribute:

    

    • For example, convert to rectangular form.
      

      

      You could stop here or figure out the trig functions:

      

  • Rectangular to Trig
    If then the magnitude is and the angle is described above.
    • Convert to trig form
      

      

      The real part is positive so the angle is

      

      so

      

    • Convert 4 to trig form
      The magnitude is 4 and the angle is 0 so

      

    • Convert to trig form
      The magnitude is 1 and the angle is so

      

• Multiplication and division in trig form
  The utility of trig form is that it makes multiplicaiton and division easy

  

  applying trig identities this is equal to

  

  So there's an easy rule for multiplying complex numbers in trig form: multiply the magnitudes and add the angles.

  Conversely, to divide you should divide the magnitudes and subtract the angles:

  

  Exponentiation is also easy, just think about what would happen if you did the multiplication rule multiple times:

  

• Back to Notes Index
  
  • Back to Notes Index
  • Current Semester Courses
  • Back to my homepage
  • Utah Math Department
  • University of Utah
  "The views, opinions and conclusions expressed in these pages are strictly those of the page author. The contents of the site have not been reviewed or approved by the University of Utah."

  Author: Christopher Cashen <nil >

  Date: 2008/12/10 15:17:50