Converting Complex Numbers
Polar to Rectangular
|Here are the formal expression for converting polar to rectangular coordinates: |
b = c(sinq)
a = c(cosq)
b is the imaginary component of the rectangular coordinate
a is the real component of the rectangular coordinate
For the sake of this lesson, lets assume that the two forms of complex numbers are expressed this way:
- Polar: cÐq
- Rectangular: a +jb
So the objective is to express cÐq in in the form of a +jb.
Recall how the same complex number can be represented two different ways:
Citing a complex number with polar coordinates.
Citing the same complex number with rectangular coordinates.
Overlay the two coordinate system to show that there is a right triangle made up of all the important parts of the complex number. Converting from polar to rectangular coordinates is a matter of solving for the two sides of the triangle in terms of its hypotenuse and angle q.
a = the length of the base of the triangle
b = the height of the triangle
c = the hypotenuse of the triangle
q = the angle
From the basic definition for the sine of an angle: b = c(sinq)
That provides the imaginary component of the rectangular form in terms of the two polar components.
From the basic definition for the cosine of an angle: a = c(cosq)
That gives us the real component of the complex number in rectangular form.
Converting from polar to rectangular coordinates for any complex number is a matter of solving those two equations.
Convert 15.4Ð29° to rectangular form
Step 1. Identify the components
c = 15.4
q = 29°
Step 2. Solve for the imaginary component
b = c(sinq) = 15.4(sin 29°) = 7.5
Step 3. Solve for the real component
a = c(cosq) = 15.4(cos 29°) = 13.5
Step 4. Put it all together
15.4Ð29° = 13.5 +j7.5
Learn From More Examples
- 10Ð45° = 7.07 +j7.07
- 10Ð-45° = 7.07 -j7.07
- 3.16Ð71.6° = 1 + j3
- 20.1Ð84.3 = 2 + j20
- 20.1Ð5.7° = 20 +j2
- 10Ð0° = 10
- 10Ð90° = j10
- 5.8Ð59° = 3 +j5
- 5.4Ð21.8° = 5 +j2
- 10.4Ð-73.3° = 3 -j10