Rucete ✏ AP Precalculus In a Nutshell
9. Polar Functions — Practice Questions
This chapter introduces the concepts of polar coordinates, graphing polar functions, and converting between rectangular and polar forms.
(Multiple Choice — Click to Reveal Answer)
1. Which of the following represents the conversion from polar to rectangular coordinates?
(A) x = r sin θ, y = r cos θ
(B) x = r cos θ, y = r sin θ
(C) x = r tan θ, y = r sin θ
(D) x = r cos θ, y = r cot θ
Answer
(B) — Polar to rectangular conversions use x = r cos θ and y = r sin θ.
2. Which of the following represents the conversion from rectangular to polar coordinates?
(A) r = √(x² + y²), θ = tan⁻¹(y/x)
(B) r = x + y, θ = sin⁻¹(y/x)
(C) r = |x − y|, θ = cos⁻¹(x/r)
(D) r = x² − y², θ = tan(x/y)
Answer
(A) — The polar form is r = √(x² + y²), θ = tan⁻¹(y/x).
3. The polar coordinate (3, π/2) corresponds to which rectangular point?
(A) (3, 0)
(B) (0, 3)
(C) (−3, 0)
(D) (0, −3)
Answer
(B) — x = r cosθ = 3·0 = 0, y = r sinθ = 3·1 = 3 ⇒ (0, 3).
4. Which of the following points corresponds to r = −4, θ = π/6?
(A) (2√3, −2)
(B) (−2√3, 2)
(C) (−2√3, −2)
(D) (2√3, 2)
Answer
(C) — x = r cosθ = −4·(√3/2) = −2√3, y = r sinθ = −4·(1/2) = −2.
5. The point (−1, 0) in rectangular coordinates can be written as which polar coordinate?
(A) (1, π)
(B) (−1, π/2)
(C) (1, 0)
(D) (1, 2π)
Answer
(A) — On the negative x-axis, r = 1 and θ = π.
6. Which complex number corresponds to the polar form 4(cos π/3 + i sin π/3)?
(A) 2 + 2√3 i
(B) 2√3 + 2 i
(C) 1 + 3 i
(D) 2 + 3 i
Answer
(A) — 4cos(π/3) = 2, 4sin(π/3) = 2√3 ⇒ 2 + 2√3 i.
7. Which polar equation represents a circle centered at the origin?
(A) r = 3
(B) r = 3 + 2cosθ
(C) r = 2sinθ
(D) r = 4cosθ
Answer
(A) — r = constant represents a circle centered at the origin.
8. The graph of r = 6cosθ is:
(A) A cardioid
(B) A circle
(C) A straight line
(D) A spiral
Answer
(B) — r = 6cosθ is a circle with diameter 6.
9. The polar equation r = 2 + 2sinθ represents:
(A) A line
(B) A parabola
(C) A cardioid
(D) A spiral
Answer
(C) — r = a + a sinθ is a cardioid.
10. Which polar equation represents a limaçon?
(A) r = 3 + cosθ
(B) r = 3
(C) r = θ
(D) r = 4sinθ
Answer
(A) — r = a + bcosθ or a + bsinθ with a ≠ b represents a limaçon.
11. Which of the following shows the correct relation between r, x, and y?
(A) r² = x² − y²
(B) r² = x² + y²
(C) r² = x + y
(D) r² = 2xy
Answer
(B) — Derived from Pythagorean theorem: r² = x² + y².
12. In polar form, what is the modulus of the complex number 1 − i?
(A) 1
(B) √2
(C) 2
(D) 3
Answer
(B) — |1 − i| = √(1² + (−1)²) = √2.
13. Which represents the argument (angle) of 1 − i?
(A) π/4
(B) 3π/4
(C) −π/4
(D) π/3
Answer
(C) — Since in Quadrant IV, θ = −π/4.
14. What is the polar form of the complex number −5i?
(A) 5(cos π + i sin π)
(B) 5(cos π/2 + i sin π/2)
(C) 5(cos 3π/2 + i sin 3π/2)
(D) 5(cos 0 + i sin 0)
Answer
(C) — −5i lies on the negative imaginary axis, θ = 3π/2.
15. Which complex number cannot be written in polar form?
(A) 1
(B) i
(C) 0
(D) 1 + i
Answer
(C) — r = 0 gives no defined angle.
16. Which represents the graph of r = 2 + 2cosθ?
(A) Circle
(B) Cardioid
(C) Spiral
(D) Ellipse
Answer
(B) — r = a + a cosθ forms a cardioid.
17. Which of the following is the rectangular form of r = 4cosθ?
(A) x² + y² = 4x
(B) x² + y² = 2x
(C) x² + y² = 8x
(D) x² + y² = 16x
Answer
(A) — Multiply both sides by r: r² = 4r cosθ → x² + y² = 4x.
18. What is the equation of a line in polar coordinates that passes through the pole?
(A) r = 3
(B) θ = π/4
(C) r = 1 + cosθ
(D) r = 2sinθ
Answer
(B) — θ = constant represents a straight line through the pole.
19. The graph of r = θ represents:
(A) A cardioid
(B) A spiral
(C) A circle
(D) A line
Answer
(B) — As θ increases, r increases proportionally → a spiral.
20. Which of the following is equivalent to the polar equation r = 3sinθ?
(A) x² + y² = 3x
(B) x² + y² = 3y
(C) x² + y² = 2x
(D) x² + y² = 2y
Answer
(B) — Multiply both sides by r: r² = 3r sinθ → x² + y² = 3y.
21. Which of the following depicts a limaçon with an inner loop?
(A) r = 2 + 3cosθ
(B) r = 3 + 2cosθ
(C) r = 2 + 2cosθ
(D) r = 1 + 2cosθ
Answer
(D) — a/b < 1 → limaçon with inner loop.
22. What happens if r < 0 in polar coordinates?
(A) The point is reflected across the pole.
(B) The radius becomes undefined.
(C) The graph disappears.
(D) The point moves along the same direction.
Answer
(A) — Negative r reflects the point across the origin.
23. The polar function r = 1 + sinθ has its minimum value at:
(A) θ = π/2
(B) θ = 0
(C) θ = 3π/2
(D) θ = π
Answer
(C) — Minimum occurs when sinθ = −1 → θ = 3π/2.
24. Which of the following is a rose curve?
(A) r = 2 + cosθ
(B) r = 2cos3θ
(C) r = 3
(D) r = θ
Answer
(B) — r = a cos(nθ) or r = a sin(nθ) → rose curve.
25. For r = 2sinθ, what is the maximum radius?
(A) 0
(B) 1
(C) 2
(D) 3
Answer
(C) — Since sinθ ≤ 1, max r = 2.
26. Which rectangular equation is equivalent to the polar curve r = 5/(1 − cosθ)?
(A) y² = 10(x + 2.5)
(B) y² = 10x
(C) (x − 5)² + y² = 25
(D) x² + y² = 5x
Answer
(A) — Using r = √(x² + y²) and cosθ = x/r: r = 5/(1 − x/r) ⇒ √(x² + y²) − x = 5. Squaring and simplifying gives y² = 10x + 25 = 10(x + 2.5).
27. Which statement about the limaçon r = 2 + 5cosθ is true?
(A) It is a cardioid touching the origin.
(B) It has an inner loop because |b| > |a|.
(C) Its maximum radius occurs at θ = π.
(D) It is symmetric about the y-axis.
Answer
(B) — For r = a + bcosθ, if |b| > |a| (5 > 2), there is an inner loop. Max radius occurs at θ = 0 (not π), and cosine symmetry is about the x-axis.
28. For the rose r = 4 sin(3θ), how many petals does the graph have and where are petal tips for 0 ≤ θ < 2π?
(A) 3 petals; at θ = kπ/3
(B) 3 petals; at θ = π/6, π/2, 5π/6
(C) 6 petals; at θ = kπ/3
(D) 3 petals; at θ = (2k+1)π/6
Answer
(D) — For r = a sin(nθ) with n odd, there are n petals. Here n = 3 → 3 petals. Petal tips when sin(3θ) = 1 → 3θ = π/2 + 2kπ ⇒ θ = (2k+1)π/6.
29. Convert the complex number z = −2 + 2√3 i to polar form r(cosθ + i sinθ) with 0 ≤ θ < 2π.
(A) 4(cos(2π/3) + i sin(2π/3))
(B) 4(cos(π/3) + i sin(π/3))
(C) 4(cos(5π/6) + i sin(5π/6))
(D) 4(cos(2π/3) − i sin(2π/3))
Answer
(A) — r = √[(-2)² + (2√3)²] = √(4 + 12) = 4. tanθ = (2√3)/(−2) = −√3 with x < 0, y > 0 → Quadrant II ⇒ θ = 2π/3.
30. Let r = 6cosθ. The average rate of change of r on [0, π/2] is:
(A) −12/π per radian
(B) −6/π per radian
(C) −3/π per radian
(D) 0
Answer
(A) — r(0) = 6, r(π/2) = 0. Average rate = (0 − 6)/(π/2 − 0) = −6 / (π/2) = −12/π.
31. Which rectangular equation corresponds to the polar curve r = 8 sinθ?
(A) x² + y² = 8x
(B) x² + y² = 8y
(C) (x − 4)² + y² = 16
(D) (y − 4)² + x² = 16
Answer
(B) — Multiply both sides by r: r² = 8r sinθ ⇒ x² + y² = 8y.
32. For r = 3 + 3 cosθ, which is true?
(A) It has an inner loop because |b| > |a|.
(B) It is a circle of radius 3 centered at (3,0).
(C) It is a cardioid (a = b) with cusp at the origin.
(D) It is symmetric about the y-axis.
Answer
(C) — a = b = 3 ⇒ cardioid; for cosine-form cardioid, cusp at the origin and symmetry about the x-axis.
33. The curve given by r = 2/(1 + sinθ) is a conic. Its focus-directrix form implies which conic and eccentricity e?
(A) Parabola, e = 1
(B) Ellipse, e < 1
(C) Hyperbola, e > 1
(D) Circle, e = 0
Answer
(A) — General conic in polar with focus at origin: r = ℓ/(1 + e sinθ). Here denominator matches 1 + e sinθ with ℓ = 2 and e = 1 ⇒ parabola.
34. The rectangular point (−3, −3√3) corresponds to which polar coordinate (r, θ) with r > 0 and 0 ≤ θ < 2π?
(A) (6, 4π/3)
(B) (6, 5π/3)
(C) (6, 2π/3)
(D) (6, 7π/6)
Answer
(A) — r = √[9 + 27] = 6. tanθ = (−3√3)/(−3) = √3 with x < 0, y < 0 ⇒ Quadrant III, principal angle π/3 in QI ⇒ θ = 4π/3.
35. For r = 4 − 6cosθ, which statement is correct?
(A) Symmetric about the y-axis; inner loop exists.
(B) Symmetric about the x-axis; inner loop exists.
(C) Symmetric about the origin; no inner loop.
(D) Not symmetric; cardioid.
Answer
(B) — Cosine-form limaçon is symmetric about the x-axis. Since |b| > |a| (6 > 4), an inner loop occurs.
36. (Short Answer) Convert the polar equation r = 10 cosθ to rectangular form.
Answer
x² + y² = 10x — Multiply by r: r² = 10r cosθ ⇒ x² + y² = 10x.
37. (Short Answer) Find the polar coordinates (r, θ) with r > 0, 0 ≤ θ < 2π for the point (x, y) = (−4, 4√3).
Answer
(8, 2π/3) — r = √(16 + 48) = 8. tanθ = (4√3)/(−4) = −√3 with x < 0, y > 0 ⇒ Quadrant II → θ = 2π/3.
38. (Short Answer) For r = 2 + 4 sinθ, compute the average rate of change of r on [π/2, π].
Answer
−8/π per radian — r(π/2) = 6, r(π) = 2 ⇒ (2 − 6)/(π − π/2) = −4/(π/2) = −8/π.
39. (Short Answer) Write z = −3 − 3i in polar form r(cosθ + i sinθ) with 0 ≤ θ < 2π.
Answer
3√2 [cos(5π/4) + i sin(5π/4)] — r=√(9+9)=3√2, angle in QIII with tanθ=1 ⇒ θ=5π/4.
40. (Short Answer) Determine whether r = 3 − 5 sinθ has an inner loop and state the symmetry axis.
Answer
Inner loop present; symmetry about the y-axis — Sine-form; |b| > |a| ⇒ inner loop. Sine gives symmetry about the line θ = π/2 (y-axis).
41. (Short Answer) Find all θ in [0, 2π) where r = 2 + 2cosθ attains its minimum radius, and state that minimum.
Answer
θ = π, r_min = 0 — r_min at cosθ = −1 ⇒ θ=π; r=2+2(−1)=0.
42. (Short Answer) Convert the polar point (r, θ) = (−5, −π/6) to rectangular coordinates.
Answer
(−(5√3)/2, 5/2) — x = r cosθ = −5·(√3/2) = −(5√3)/2; y = r sinθ = −5·(−1/2) = 5/2.
43. (Short Answer) For r = 4 cosθ, find two distinct polar representations of the same point at θ = π/3 with r ≥ 0 and r < 0, respectively.
Answer
(2, π/3) and (−2, 4π/3) — r(π/3)=4·(1/2)=2. Negative-r representation adds π to θ: (−2, π/3+π)=(−2, 4π/3).
44. (Short Answer) Determine the number of petals and petal length for r = 5 cos(4θ).
Answer
8 petals, length 5 — For r = a cos(nθ) with n even, petals = 2n = 8, each reaches r = a = 5.
45. (Short Answer) Convert z = 8(cos(π/6) + i sin(π/6)) to rectangular form.
Answer
4√3 + 4i — 8·cos(π/6)=8·(√3/2)=4√3; 8·sin(π/6)=8·(1/2)=4.
46. (Short Answer) For r = 3/(1 − sinθ), classify the conic and identify e.
Answer
Parabola, e = 1 — Matches r = ℓ/(1 − e sinθ) with e = 1.
47. (Short Answer) The curve r = 7 is intersected by the curve r = 14 cosθ. Find all intersection angles in [0, 2π).
Answer
θ = π/3, 5π/3 — Set 7 = 14 cosθ ⇒ cosθ = 1/2 ⇒ θ = ±π/3 mod 2π → π/3, 5π/3.
48. (Short Answer) Evaluate the average rate of change of r = 1 + cos(2θ) on [0, π].
Answer
0 — r(0) = 2, r(π) = 2 ⇒ (2 − 2)/(π − 0) = 0.
49. (Short Answer) Convert the polar equation r = 6/(2 + cosθ) into a rectangular equation.
Answer
3x² + 4y² + 12x − 36 = 0 — From r(2 + cosθ) = 6 ⇒ 2r + x = 6. Replacing r with √(x² + y²): 2√(x² + y²) = 6 − x → squaring gives 4(x² + y²) = (6 − x)² ⇒ 3x² + 4y² + 12x − 36 = 0.
50. (Short Answer) For r = 2 − 2 sinθ, find the minimum radius and the angle(s) where it occurs.
Answer
r_min = 0 at θ = π/2 — Minimum when sinθ = 1 → θ = π/2 ⇒ r = 2 − 2·1 = 0.