Converting Complex Numbers Polar to Rectangular Here are the formal expression for converting polar to rectangular coordinates: b = c(sinq) a = c(cosq) where: 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. Example 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
 10Ð45° = 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
 10.4Ð73.3° = 3 j10
