SME
Grade 12 Math · Problem type
Multiplying in polar form
Use this for “Multiply 2(cos 30° + i sin 30°) by 3(cos 60° + i sin 60°).”
Diagram
z = 2(cos 30° + i sin 30°) ≈ 1.73 + i.
z × w = (1.73 + i)(1.5 + 2.6i)
z × w = 6i
The moduli multiply, 2 × 3 = 6, and the arguments add, 30° + 60° = 90°.
This is an example. Type your own values to start a new problem.
r₁
(cos
θ₁
°
+
i
sin
θ₁
°)
×
r₂
(cos
θ₂
°
+
i
sin
θ₂
°)
=
r
(cos
θ
°
+
i
sin
θ
°)
p
Real part of zw
worked out (not typed)
q
Imaginary part of zw
worked out (not typed)
Formulas
r
=
r₁
×
r₂
6 = 2 × 3
θ
=
θ₁
+
θ₂
90 = 30 + 60
p
=
r
×
cos(
θ
°)
0 = 6 × cos(90°)
q
=
r
×
sin(
θ
°)
6 = 6 × sin(90°)
Assumptions
•
Multiply the moduli and add the arguments.
•
So multiplying by w stretches z by |w| and turns it by w’s argument.
•
Subtract 360° from an argument past 360° if you want it on one turn.
Step-by-step
WE KNOW
r₁ = 2, θ₁ = 30°, r₂ = 3, θ₂ = 60°
FIND
modulus of zw (r), argument of zw (θ), real part of zw (p), imaginary part of zw (q)
Step 1 · Find modulus of zw (r)
Use r = r₁ × r₂
Moduli multiply.
r
=
2
×
3
r
=
6
Step 2 · Find argument of zw (θ)
Use θ = θ₁ + θ₂
Arguments add: the turns add up.
θ
=
θ₁
°
+
θ₂
°
θ
=
30°
+
60°
θ
=
90°
Step 3 · Find real part of zw (p)
Use p = r × cos(θ°)
Back
to
a
+
bi:
the
real
part
of
zw
is
r
cos
θ
.
p
=
6
×
cos(90°)
p
=
6
×
0
p
=
0
Step 4 · Find imaginary part of zw (q)
Use q = r × sin(θ°)
The
imaginary
part
of
zw
is
r
sin
θ
.
q
=
6
×
sin(90°)
q
=
6
Check
Put the numbers back in. Both sides should match.
6 = 2 × 3 ✓
90 = 30 + 60 ✓
0 = 6 × cos(90°) ✓
6 = 6 × sin(90°) ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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Related lessons
Polar coordinates and polar form of complex numbers
Main lesson
Polar form of a complex number
Use this for “Write 2(cos 60° + i sin 60°) as a + bi,” or back.
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Use this for “Find (1 + i)⁸ with De Moivre’s theorem.”
The nth roots of a complex number
Use this for “Find the cube roots of 8i” or “the fourth roots of −16.”
Rose curves
Use this for “How many petals does r = 4 cos 2θ have? Find r at θ = 30°.”
Limaçons and cardioids
Use this for “Find r on r = 2 + 2 cos θ at θ = 60°” and the curve’s shape.
Circles through the pole
Use this for “Graph r = 6 cos θ: find r at θ = 60°, the center and the radius.”
The spiral r = aθ
Use this for “Graph r = 0.5θ: find r at θ = 270°” or “At what angle is r = 6?”