Rucete ✏ AP Precalculus In a Nutshell
9. Polar Functions — Practice Questions 3
This chapter explores how polar equations describe curves, how to convert between rectangular and polar forms, and how to analyze key features of standard polar graphs.
(Multiple Choice — Click to Reveal Answer)
1. Which identity correctly relates rectangular and polar coordinates?
(A) x = r sinθ, y = r cosθ
(B) x = r cosθ, y = r sinθ
(C) x = r tanθ, y = r cotθ
(D) x = r, y = θ
Answer
(B) — The standard relations are x = r cosθ and y = r sinθ.
2. Which identity is always true for r, x, and y?
(A) r = x + y
(B) r² = x² + y²
(C) r = x² + y²
(D) r² = x + y
Answer
(B) — From the right triangle with legs x and y: r² = x² + y².
3. Convert (r, θ) = (5, π/6) to rectangular coordinates.
(A) (5√3/2, 5/2)
(B) (5/2, 5√3/2)
(C) (−5√3/2, 5/2)
(D) (5√3/2, −5/2)
Answer
(A) — x = 5cos(π/6) = 5√3/2, y = 5sin(π/6) = 5/2.
4. Convert (x, y) = (−4, 0) to polar coordinates with r > 0 and 0 ≤ θ < 2π.
(A) (4, 0)
(B) (4, π)
(C) (−4, 0)
(D) (4, 2π)
Answer
(B) — On the negative x-axis: r = 4, θ = π.
5. Which graph is a circle centered at the origin in polar form?
(A) r = 4
(B) r = 4cosθ
(C) r = 4sinθ
(D) r = 4 + cosθ
Answer
(A) — r = constant is a circle of radius 4 centered at the origin.
6. Which polar curve is a circle centered on the positive x-axis?
(A) r = 6
(B) r = 6cosθ
(C) r = 6sinθ
(D) r = θ
Answer
(B) — r = a cosθ is a circle centered at (a/2, 0) with radius a/2.
7. For r = 3 + 3sinθ, the curve is a:
(A) rose
(B) spiral
(C) cardioid
(D) circle
Answer
(C) — a = b ⇒ cardioid.
8. Which equation represents a limaçon with inner loop?
(A) r = 4 + 2cosθ
(B) r = 2 + 4sinθ
(C) r = 3 + 3cosθ
(D) r = 5
Answer
(B) — |b| > |a| (4 > 2) ⇒ inner loop.
9. Which polar equation represents a rose curve?
(A) r = 2 + cosθ
(B) r = 4sin(2θ)
(C) r = 3
(D) r = θ
Answer
(B) — r = a sin(nθ) or a cos(nθ) gives a rose (n even ⇒ 2n petals).
10. Which statement about negative r is true?
(A) The point is reflected through the origin.
(B) The point is unchanged.
(C) The angle θ is halved.
(D) The point moves to the positive y-axis.
Answer
(A) — Negative r flips the point across the origin.
11. Convert z = 2 + 2i to polar form r(cosθ + i sinθ) with 0 ≤ θ < 2π.
(A) 2√2[cos(π/4) + i sin(π/4)]
(B) 2√2[cos(3π/4) + i sin(3π/4)]
(C) 2[cos(π/2) + i sin(π/2)]
(D) √8[cos(π/6) + i sin(π/6)]
Answer
(A) — r = √(2² + 2²) = 2√2, θ = π/4 (Q I).
12. Which complex number has modulus 5 and argument π?
(A) −5
(B) 5
(C) 5i
(D) −5i
Answer
(A) — r = 5, θ = π ⇒ −5 + 0i.
13. Which identity converts from rectangular to polar angle?
(A) θ = tan⁻¹(y/x)
(B) θ = sin⁻¹(x/r)
(C) θ = cos⁻¹(y/r)
(D) θ = tan(x/y)
Answer
(A) — With quadrant adjustments, θ = atan2(y, x); in simple form tan⁻¹(y/x).
14. The maximum radius of r = 5cosθ is:
(A) 0
(B) 2.5
(C) 5
(D) 10
Answer
(C) — Max when cosθ = 1 ⇒ r = 5.
15. The curve r = 2 + 2cosθ passes through the origin at:
(A) θ = 0
(B) θ = π/2
(C) θ = π
(D) θ = 3π/2
Answer
(C) — r = 2 + 2cosθ = 0 ⇒ cosθ = −1 ⇒ θ = π.
16. Which polar equation is symmetric about the y-axis?
(A) r = a cosθ
(B) r = a sinθ
(C) r = a + cosθ
(D) r = θ
Answer
(B) — Sine-based forms are symmetric about θ = π/2 (y-axis).
17. Convert r = 3sinθ to rectangular form.
(A) x² + y² = 3x
(B) x² + y² = 3y
(C) x² + y² = 6x
(D) x² + y² = 6y
Answer
(B) — Multiply by r: r² = 3r sinθ ⇒ x² + y² = 3y.
18. Convert r = 8cosθ to rectangular form.
(A) x² + y² = 8y
(B) x² + y² = 8x
(C) (x − 4)² + y² = 16
(D) (y − 4)² + x² = 16
Answer
(B) — r² = 8r cosθ ⇒ x² + y² = 8x.
19. The rose r = 2cos(3θ) has how many petals, and each petal’s maximum length is:
(A) 3 petals; length 2
(B) 6 petals; length 2
(C) 3 petals; length 1
(D) 6 petals; length 1
Answer
(A) — Odd n = 3 ⇒ 3 petals; amplitude a = 2 is max radius.
20. Which is a conic in polar form?
(A) r = 4 + sinθ
(B) r = 3/(1 − cosθ)
(C) r = 3cos(2θ)
(D) r = θ
Answer
(B) — r = ℓ/(1 ± e cosθ) or ℓ/(1 ± e sinθ) is a conic.
21. For r = 1 + 2cosθ, which is true?
(A) Cardioid
(B) Limaçon with inner loop
(C) Circle
(D) Line
Answer
(B) — |b| > |a| ⇒ inner loop.
22. The curve r = θ (θ ≥ 0) is a(n):
(A) limaçon
(B) cardioid
(C) Archimedean spiral
(D) rose
Answer
(C) — r proportional to θ is an Archimedean spiral.
23. Which angle produces the minimum r for r = 3 + 2sinθ?
(A) θ = 0
(B) θ = π/2
(C) θ = π
(D) θ = 3π/2
Answer
(D) — sinθ minimal at −1 ⇒ θ = 3π/2 ⇒ r_min = 1.
24. Convert z = −4i to polar form r(cosθ + i sinθ) with 0 ≤ θ < 2π.
(A) 4[cos(π/2) + i sin(π/2)]
(B) 4[cos(3π/2) + i sin(3π/2)]
(C) 4[cosπ + i sinπ]
(D) 4[cos0 + i sin0]
Answer
(B) — −4i lies on negative imaginary axis ⇒ θ = 3π/2.
25. For r = 2 + 4cosθ on [0, π], the average rate of change is:
(A) −8/π
(B) −4/π
(C) −2/π
(D) 0
Answer
(A) — r(π) = −2, r(0) = 6 ⇒ (−2 − 6)/(π − 0) = −8/π.
26. Convert (x, y) = (−3, 3√3) to polar coordinates with r > 0 and 0 ≤ θ < 2π.
(A) (6, 2π/3)
(B) (6, π/3)
(C) (3√2, 5π/4)
(D) (6, 4π/3)
Answer
(A) — r = √(9 + 27) = 6; tanθ = −√3 with x < 0, y > 0 ⇒ QII ⇒ θ = 2π/3.
27. Which rectangular equation corresponds to r = 10/(1 + sinθ)?
(A) x² + y² = 10x
(B) (x − 5)² + y² = 25
(C) y² = 10(x + 5)
(D) x² + y² + y√(x² + y²) = 10
Answer
(D) — r(1 + sinθ) = 10 ⇒ r + r sinθ = 10 ⇒ √(x² + y²) + y = 10 ⇒ x² + y² + y√(x² + y²) = 10.
28. For r = 2 + 5cosθ, which feature occurs?
(A) Cardioid
(B) Inner loop
(C) Circle
(D) Rose
Answer
(B) — |b| > |a| (5 > 2) ⇒ inner loop.
29. The rose r = 3sin(4θ) has:
(A) 4 petals, each of length 3
(B) 8 petals, each of length 3
(C) 4 petals, each of length 6
(D) 8 petals, each of length 6
Answer
(B) — Even n = 4 ⇒ 2n = 8 petals; amplitude a = 3 is max radius.
30. Which statement is true for r = 4cosθ?
(A) It is symmetric about the y-axis.
(B) It is symmetric about the x-axis.
(C) It is symmetric about the line θ = π/4.
(D) It passes through the origin.
Answer
(D) — r = 0 at θ = π/2 ⇒ the curve passes through the origin; cosine-form is symmetric about the x-axis, but the key property asked is passage through the origin.
31. Convert z = −2 − 2√3 i to polar form with 0 ≤ θ < 2π.
(A) 4[cos(4π/3) + i sin(4π/3)]
(B) 4[cos(2π/3) + i sin(2π/3)]
(C) 2[cos(5π/6) + i sin(5π/6)]
(D) 4[cos(5π/6) + i sin(5π/6)]
Answer
(A) — r = √(4 + 12) = 4; x < 0, y < 0 ⇒ QIII; reference angle π/3 ⇒ θ = 4π/3.
32. Find the average rate of change of r = 5 − 3sinθ on [π/6, 5π/6].
(A) −6/π
(B) 0
(C) 6/π
(D) −3/π
Answer
(B) — r(π/6) = 5 − 3·(1/2) = 3.5; r(5π/6) = 5 − 3·(1/2) = 3.5 ⇒ average rate 0.
33. Which polar equation represents a line through the pole?
(A) r = 3
(B) θ = 2π/3
(C) r = 1 + cosθ
(D) r = 1 + 2sinθ
Answer
(B) — θ = constant is a line through the origin.
34. Which is symmetric about the origin?
(A) r = a sin(2θ)
(B) r = a + cosθ
(C) r = a sinθ
(D) r = a cosθ
Answer
(A) — Double-angle sine exhibits origin symmetry (r(θ + π) = −r(θ)).
35. Convert the circle x² + y² + 6x = 0 to polar form.
(A) r = 3cosθ
(B) r = −3cosθ
(C) r = 6cosθ
(D) r = 3sinθ
Answer
(B) — r² = −6x ⇒ r² = −6r cosθ ⇒ r = −3cosθ (for the standard r ≥ 0 representation, this traces via negative r).
36. Convert (x, y) = (4, −4√3) to polar coordinates (r, θ) with r > 0 and 0 ≤ θ < 2π.
Answer
(8, 5π/3) — r = √(16 + 48) = 8; tanθ = (−4√3)/4 = −√3 with x > 0, y < 0 ⇒ QIV ⇒ θ = 5π/3.
37. Give another polar representation of the same point as (r, θ) = (7, π/8) using negative r.
Answer
(−7, 9π/8) — Add π to θ and negate r to represent the same point.
38. Convert r = 9/(2 + cosθ) to a rectangular equation with no radicals.
Answer
3x² + 4y² + 12x − 81 = 0 — r(2 + cosθ) = 9 ⇒ 2r + x = 9 ⇒ 2√(x² + y²) = 9 − x; square and simplify ⇒ 4(x² + y²) = (9 − x)² ⇒ 3x² + 4y² + 12x − 81 = 0.
39. State the angles (0 ≤ θ < 2π) where r = 2 + 3sinθ attains its minimum and maximum values, and those values.
Answer
Min: θ = 3π/2, r = −1; Max: θ = π/2, r = 5 — sinθ ∈ [−1, 1].
40. Convert z = −1 + √3 i to polar form r(cosθ + i sinθ) with 0 ≤ θ < 2π.
Answer
2[cos(2π/3) + i sin(2π/3)] — r = √(1 + 3) = 2; QII with reference π/3 ⇒ θ = 2π/3.
41. Find all intersection angles (0 ≤ θ < 2π) of r = 5 and r = 10cosθ.
Answer
θ = π/3, 5π/3 — Set 5 = 10cosθ ⇒ cosθ = 1/2.
42. Convert the polar equation r = 6sinθ to a circle in standard rectangular form (complete the square if needed).
Answer
x² + (y − 3)² = 9 — r² = 6r sinθ ⇒ x² + y² = 6y ⇒ x² + (y − 3)² = 9.
43. For r = 1 + cosθ, list the intervals of θ on [0, 2π] where r is increasing and where r is decreasing.
Answer
Increasing on (π, 2π); decreasing on (0, π) — dr/dθ = −sinθ; positive on (π, 2π), negative on (0, π); endpoints are extrema.
44. Determine whether the point given by (−3, −π/6) equals the point (3, 5π/6) in polar coordinates. Justify.
Answer
Equal — (−3, −π/6) = (3, −π/6 + π) = (3, 5π/6).
45. Convert (r, θ) = (−6, 2π/3) to rectangular coordinates.
Answer
(3, −3√3) — x = −6·cos(2π/3) = −6·(−1/2) = 3; y = −6·sin(2π/3) = −6·(√3/2) = −3√3.
46. For r = 4 − 4cosθ, state whether an inner loop occurs and give the axis of symmetry.
Answer
Inner loop present; symmetric about the x-axis — |b| = |a| gives cusp at origin if equal, but here r_min = 0 and r becomes negative; cosine form is symmetric about x-axis; since at θ = 0, r = 0 and for small θ > 0, r < 0, an inner loop forms.
47. Find the average rate of change of r = 3 + 2cosθ on [π/2, 3π/2].
Answer
0 — r(π/2) = 3, r(3π/2) = 3 ⇒ (3 − 3)/(π) = 0.
48. Convert r = 2/(1 − cosθ) to a rectangular equation and name the conic.
Answer
y² = 4(x + 1) — r(1 − cosθ) = 2 ⇒ r − x = 2 ⇒ √(x² + y²) = x + 2; square: x² + y² = x² + 4x + 4 ⇒ y² = 4x + 4 ⇒ a parabola opening right.
49. Convert (x, y) = (−√3, −1) to polar coordinates with r > 0 and 0 ≤ θ < 2π.
Answer
(2, 7π/6) — r = √(3 + 1) = 2; tanθ = (−1)/(−√3) = 1/√3, with x < 0, y < 0 ⇒ QIII ⇒ θ = π + π/6 = 7π/6.
50. The rose r = 6cos(2θ) achieves maximum radius at which angles in [0, 2π)? List all.
Answer
θ = 0, π/2, π, 3π/2 — Max when cos(2θ) = 1 ⇒ 2θ = 2kπ ⇒ θ = kπ with k = 0, 1; also periodicity gives every π/2 due to 2θ, so θ = 0, π, and adding π/2 gives π/2, 3π/2.