SME
Grade 12 Math · Precalculus
Polar coordinates and polar form of complex numbers
Use this for “Convert the polar point (4, 150°) to rectangular coordinates,” or a rectangular point to polar.
Diagram
(r, θ) = (4, 150°)
x = r cos θ ≈ −3.46
y = r sin θ ≈ 2
This is an example. Type your own values to start a new problem.
x
=
r
cos
θ
°
y
=
r
sin
θ
°
Formulas
x
=
r
×
cos(
θ
°)
−2
√
3
=
4
×
cos(150°)
y
=
r
×
sin(
θ
°)
2 = 4 × sin(150°)
r
2
=
x
2
+
y
2
4
2
=
(−2
√
3
)
2
+
2
2
tan
θ
=
y
x
,
θ
in
the
quadrant
of
(
x
,
y
),
turned
180°
when
r
<
0
tan
150
=
2
−2
√
3
,
150
in
the
quadrant
of
(−2
√
3
,
2),
turned
180°
when
4
<
0
Assumptions
•
θ is measured from the positive x-axis, counterclockwise.
•
A point has many polar names: add 360° to θ, or make r negative and add 180°.
•
A negative r lands on the ray opposite θ.
Step-by-step
WE KNOW
r = 4, θ = 150°
FIND
x-coordinate (x), y-coordinate (y)
Step 1 · Find x-coordinate (x)
Use x = r × cos(θ°)
Go
r
along
the
angle;
the
across
part
is
r
cos
θ
.
x
=
4
×
cos(150°)
x
=
4
×
(−
√
3
2
)
x
=
−2
√
3
≈
−3.4641
Step 2 · Find y-coordinate (y)
Use y = r × sin(θ°)
The
up
part
is
r
sin
θ
.
y
=
4
×
sin(150°)
y
=
4
×
0.5
y
=
2
Check
Put the numbers back in. Both sides should match.
−2
√
3
=
4
×
cos(150°)
✓
2 = 4 × sin(150°) ✓
4
2
=
(−2
√
3
)
2
+
2
2
✓
tan(150°)
=
2
−3.4641
✓
Topic check
See what you know: 5 questions on this topic
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More problem types
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 “Multiply 2(cos 30° + i sin 30°) by 3(cos 60° + i sin 60°).”
Powers by De Moivre’s theorem
Use this for “Find (1 + i)⁸ with De Moivre’s theorem.”
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Use this for “Find the cube roots of 8i” or “the fourth roots of −16.”
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Use this for “How many petals does r = 4 cos 2θ have? Find r at θ = 30°.”
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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?”