SME
Grade 12 Math · Problem type
Polar form of a complex number
Use this for “Write 2(cos 60° + i sin 60°) as a + bi,” or back.
Diagram
z = 2(cos 60° + i sin 60°) ≈ 1 + 1.73i.
Real part: 1
Imaginary part: √3
This is an example. Type your own values to start a new problem.
r
(cos
θ
°
+
i
sin
θ
°)
=
a
+
b
i
Formulas
a
=
r
×
cos(
θ
°)
1 = 2 × cos(60°)
b
=
r
×
sin(
θ
°)
√
3
=
2
×
sin(60°)
r
=
√
a
2
+
b
2
2
=
√
1
2
+
√
3
²
tan
θ
=
b
a
,
θ
in
the
quadrant
of
(
a
,
b
)
tan
60
=
√
3
1,
60
in
the
quadrant
of
(1,
√
3
)
Assumptions
•
r is the modulus |z| and θ the argument, measured from the positive real axis.
•
a = r cos θ is the real part and b = r sin θ the imaginary part.
•
Going back, r = √(a² + b²) and θ is in the quadrant of the point (a, b).
Step-by-step
WE KNOW
r = 2, θ = 60°
FIND
real part (a), imaginary part (b)
Step 1 · Find real part (a)
Use a = r × cos(θ°)
The real part is the modulus times the cosine of the argument.
a
=
2
×
cos(60°)
a
=
2
×
0.5
a
=
1
Step 2 · Find imaginary part (b)
Use b = r × sin(θ°)
The imaginary part is the modulus times the sine of the argument.
b
=
2
×
sin(60°)
b
=
2
×
√
3
2
b
=
√
3
≈
1.7321
Check
Put the numbers back in. Both sides should match.
1 = 2 × cos(60°) ✓
√
3
=
2
×
sin(60°)
✓
2
=
√
1
2
+
√
3
²
✓
tan(60°)
=
1.7321
1
✓
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Polar coordinates and polar form of complex numbers
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