2.3 Polar Coordinates
Objective: to convert rectangular
coordinates to polar and vice versa
Review Complex Plane
Imagine that you are walking to the point 2 + 3i. Instead
of walking 2 units right, turning 90⁰, and walking 3
more units, you want to take the nearest route. You
need to have a direction and a distance to walk.
What is the direction you need to take?
How far should you walk?
Polar Coordinates
Every polar coordinate has both an angle (direction) and
a radius (distance).
(r, θ)
Partner Work – each group needs a balloon, two rubber
bands, and a permanent marker. One person will be
the coach, the other is the recorder.
Plot the Points
 (4, 0⁰)
 (2, 3π/2)
 (-2, 210⁰)
 (3, 135⁰)
 Now, on paper, plot the following points:
 (4, 30o)
 (-4, 225o)
 (2, -300o)
 (-3, -270o)
Polar Form of Complex Numbers
 The polar form or trigonometric form of the complex
number a + bi is r(cos θ + i sin θ).
 To express -3 + 4i in polar form, first find the radius
and the argument.
 r = √(32 + 42)
 -3 + 4i = 5(cos 2.21 + i sin 2.21)
= 5 cis 2.21
 Express 2 – 2i√5 in polar form.
 Express 1 + √3i in polar form.
θ = tan-1(4/-3)
QII
From Polar Form to Rectangular
 a = r cos θ and b = r sin θ
 Express 5 cis π/6 in rectangular form.
 x = 5 cos 30⁰
y = 5 sin 30⁰
 x = 5 (√(3)/2)
y = 5 (1/2)
 5√3/2 + 5/2 i
 Express 10 cis 300⁰ in rectangular form.
 Express 4 cis 135o in rectangular form.
Practice Problems
 Find the magnitude and argument of each of the following
numbers.
2√3 – 2i
4 cis 300o
 If z = 5 cis 75o and w = 2 cis 100o, find |zw| and arg(zw).
 Assignment page 95 8, 9a, 10, 15