Guided Notes – Complex numbers in Rectangular and Polar Form
Standards:
 N.CN.4 Represent complex numbers on the complex plane in rectangular and polar form (including real and
imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the
same number.
 N.CN.5 Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on
the complex plane; use properties of this representation for computation.
 N.CN.6 Calculate the distance between numbers in the complex plane as the modulus of the difference, and the
midpoint of a segment as the average of the numbers at its endpoints.
Essential Question: How can we represent complex numbers in trigonometric form? Why must the quadrant of a
complex number be determined before the trigonometric form can be determined?
Words to Know
Rectangular form of a complex number – the rectangular form of a complex number is z = a + bi,
where a and b are real numbers and i is the imaginary unit, √−1.
Trignometric form of a complex number – the trigonometric, or polar, form of a complex number
𝑏
z = a + bi is z = r(cosθ + isinθ) or z = r (cisθ); a = r cosθ, b = r sinθ, r = √𝑎2 + 𝑏 2 , and tan θ = ,
𝑎
where a and b are real numbers.
An electric circuit has a resistor in series with a reactance
across a capacitor. A capacitor is a device that temporarily stores
electric current. The units of both devices can be expressed in
ohms, or Ω. A diagram of the resistor and capacitor in series is
shown.
The total effective resistance to the flow of the current, or impedance, by the resistor and the capacitor in
series can be represented using a complex number, Z. The total impedance, Z, of a current through a resistor and a
capacitor is: 𝑍 = 𝑅 − 𝑋𝑐 𝑖 where R is the ohms of resistance of the resistor, and Xc is the ohms of the capacitor.
If the resistor is 5Ω and the capacitor is 2Ω, find the complex number, in rectangular florm, that expresses the
impedance of the two devices.
1.
How do we find the impedance, Z, in rectangular form,
using 𝑍 = 𝑅 − 𝑋𝑐 𝑖?
2.
Graph the complex number, Z, and identify the quadrant.
3.
How can the rectangular form be translated into trigonometric form?
4.
How can we identify which quadrant any complex number would be graphed in?
Practice
Students will use the polar, or trigonometric, form of complex numbers. Given a complex number, z, in the
form z = a + bi, the complex number can be converted to trigonometric form, z = r(cosθ + isinθ) or z = r(cisθ)
1.
What is the complex number z = 18 + 3i in trigonometric form?
a)
First identify in which quadrant the point is located.
b)
Now determine r using r = √𝑎2 + 𝑏 2
c)
Now determine θ using tan θ = 𝑎. Use the inverse trigonometric function tan-1 to isolate θ
d)
Finally substitute r and θ into the equation r(cosθ + isinθ).
𝑏
2.
What is the complex number z = 1 - 9i in trigonometric form?
3.
What is the complex number z = -10 + 2i in trigonometric form?
4.
What is the complex number z = -5 - i in trigonometric form?