Polar Coordinates — Challenge
Distance and direction describe a point just as well as across and up.
What these problems cover
These questions use 7 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Radius — Centre to edge. Half the diameter.
- Parabola — The U-shaped graph of a quadratic function.
- Ray — One endpoint and everything beyond it in one direction. Written AB.
- Complex number — Anything of the form a + bi, with a the real part and b the imaginary part.
- Complex plane — The plane where the horizontal axis is real and the vertical axis imaginary.
- Rectangular form — The ordinary (x, y) description.
- Quadrant — One of the four regions the axes divide the plane into.
Problem 1
What is the graph of r = 2cosθ?
- A circle of radius 1 centred at (1,0)
- A circle of radius 2 centred at the origin
- A cosine wave
- A straight line
- A circle of radius 1 centred at the origin
Worked answer
A circle of radius 1 centred at (1,0)
Multiply both sides by r: r2 = 2rcosθ. Now r2 = x2+y2 and rcosθ = x, so x2+y2 = 2x. Completing the square gives (x-1)2 + y2 = 1 — a circle of radius 1 centred at (1,0). It is not a wave, however much the formula looks like one.
Problem 2
Write (-2, 2√3) in polar form with 0° ≤ θ < 360°.
- (4, 120°)
- (4, 60°)
- (4, 240°)
- (2, 120°)
- (4, 300°)
Worked answer
(4, 120°)
r = √4 + 12 = √16 = 4. tanθ = 2√3-2 = -√3, so the reference angle is 60°. The point is left and up, which is quadrant II, so θ = 180° - 60° = 120°.
Problem 3
What does r = θ trace for θ ≥ 0?
- A spiral
- A circle
- A straight line
- A parabola
- A rose
Worked answer
A spiral
As the angle grows, the distance grows with it. So each time round, the curve is further out than it was on the previous turn. That is a spiral — specifically the Archimedean spiral, whose turns are evenly spaced.
Problem 4
Multiply 2(cos 30° + isin 30°) by 3(cos 60° + isin 60°).
- 6i
- -6i
- 6
- 5i
- 36i
Worked answer
6i
In polar form, multiplying multiplies the distances and adds the angles. So the product is 6(cos 90° + i sin 90°). cos 90° = 0 and sin 90° = 1, giving 6i. Expanding the brackets instead would take four times the work.
Problem 5
What is the distance between the polar points (3, 0°) and (3, 90°)?
- 3√2
- 3
- 6
- 90
- 4.5
Worked answer
3√2
Both points are 3 from the origin, and the directions differ by 90°. So the origin and the two points form a right isosceles triangle with legs 3. The distance is √9+9 = 3√2 ≈ 4.24.
Problem 6
Which of these is not the same point as (4, 30°)?
- (-4, 30°)
- (4, 390°)
- (-4, 210°)
- (4, -330°)
- (4, 750°)
Worked answer
(-4, 30°)
Adding or subtracting whole turns changes nothing, so 390°, -330° and 750° all name the same direction as 30°. A negative r means walking backwards, so (-4, 210°) lands at 4 in the 30° direction — the same point again. But (-4, 30°) walks backwards along the 30° ray, landing on the opposite side of the origin.
Problem 7
What is the graph of r = 4sinθ?
- A circle of radius 2 centred at (0,2)
- A circle of radius 4 centred at the origin
- A sine wave
- A straight line
- A circle of radius 2 centred at (2,0)
Worked answer
A circle of radius 2 centred at (0,2)
Multiply by r: r2 = 4rsinθ, so x2+y2 = 4y. Complete the square in y: x2 + (y-2)2 = 4. That is a circle of radius 2 centred at (0,2) — it sits above the origin because the sine picks out the vertical direction.
Problem 8
For z = 3(cos 40° + isin 40°), what is |z|?
- 3
- 40
- √3
- 9
- 1
Worked answer
3
Polar form for a complex number is r(cosθ + isinθ), where r is the distance from the origin. Here r = 3. The 40° is the argument — the direction — and has nothing to do with the modulus.