Trigonometry Polar Coordinates 5 — Questions and Answers
Question 1: Find the area of the region inside r = 4 cos θ and outside r = 2.
- π/3 + √3
- 4π/3 − √3 (Correct answer)
- (4π/3) + √3
- (2π/3) + √3
Correct answer: 4π/3 − √3
The intersections are at θ = ±π/3; integrating gives area = 4π/3 − √3.
Question 2: The polar curve r = θ (for θ ≥ 0) is called a(n):
- Cardioid
- Archimedean spiral (Correct answer)
- Logarithmic spiral
- Hyperbolic spiral
Correct answer: Archimedean spiral
r = aθ defines an Archimedean spiral, where the distance between arms is constant.
Question 3: Convert r = 2/(3 − sin θ) to rectangular form and identify the conic.
- Ellipse with e = 1/3 (Correct answer)
- Hyperbola with e = 3
- Parabola
- Circle
Correct answer: Ellipse with e = 1/3
Dividing numerator and denominator by 3 gives r = (2/3)/(1 − (1/3)sin θ), so e = 1/3 < 1, an ellipse.
Question 4: At the pole, the curve r = sin θ cos θ has tangent lines. How many distinct tangent lines exist there?
- 1
- 2 (Correct answer)
- 3
- 4
Correct answer: 2
r = 0 at θ = 0, π/2, π, 3π/2, but the tangent directions at θ = 0 and π are the same line, as are θ = π/2 and 3π/2, giving 2 distinct tangent lines.
Question 5: What is the Cartesian form of r = sec θ?
- y = 1
- x = 1 (Correct answer)
- x² + y² = 1
- xy = 1
Correct answer: x = 1
r = sec θ means r = 1/cos θ → r cos θ = 1 → x = 1.
Question 6: For the polar curve r = 3 sin(2θ), how many petals are in the region θ ∈ [0, 2π)?
- 2
- 3
- 4 (Correct answer)
- 6
Correct answer: 4
r = 3 sin(2θ) is a four-petaled rose because n = 2 is even, giving 2n = 4 petals.
Question 7: If the polar coordinates of a point are (r, θ), what are the polar coordinates of the point reflected across the line θ = π/2?
- (r, π − θ) (Correct answer)
- (r, π + θ)
- (−r, θ)
- (r, 2π − θ)
Correct answer: (r, π − θ)
Reflection across θ = π/2 (the y-axis) maps (r, θ) to (r, π − θ).
Trigonometry Polar Coordinates 5
Find the area of the region inside r = 4 cos θ and outside r = 2.