SME
Grade 12 Math · Problem type
Circles through the pole
Use this for “Graph r = 6 cos θ: find r at θ = 60°, the center and the radius.”
Diagram
r = 6 cos θ: a circle of diameter 6 through the pole, θ from 0° to 180°.
(r, θ) = (3, 60°)
This is an example. Type your own values to start a new problem.
r
=
a
cos
θ
°
h
Center x-coordinate
calculated
R
Radius
worked out (not typed)
Formulas
r
=
a
×
cos(
θ
°)
3 = 6 × cos(60°)
h
=
a
2
3
=
6
2
R
=
|
a
|
÷
2
3 = |6| ÷ 2
Assumptions
•
r = a cos θ is a circle through the pole with diameter |a| along the x-axis.
•
Times r: r² = ar cos θ, so x² + y² = ax, a circle centered at (a/2, 0) with radius |a|/2.
•
The whole circle is drawn as θ runs from 0° to 180°; past 90° r is negative.
Step-by-step
WE KNOW
a = 6, θ = 60°
FIND
directed distance r (r), center x-coordinate (h), radius (R)
Step 1 · Find directed distance r (r)
Use r = a × cos(θ°)
Put
θ
into
the
curve’s
equation.
r
=
6
×
cos(60°)
r
=
6
×
0.5
r
=
3
Step 2 · Find center x-coordinate (h)
Use h = a ÷ 2
The center is halfway along the diameter from the pole, on the x-axis.
h
=
6
2
h
=
3
Step 3 · Find radius (R)
Use R = |a| ÷ 2
The radius is half the diameter.
R
=
|6|
÷
2
R
=
6
2
R
=
3
Check
Put the numbers back in. Both sides should match.
3 = 6 × cos(60°) ✓
3
=
6
2
✓
3 = |6| ÷ 2 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Refresh
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.
Multiplying in polar form
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.”
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.
The spiral r = aθ
Use this for “Graph r = 0.5θ: find r at θ = 270°” or “At what angle is r = 6?”