Trigonometry Polar Coordinates 3 — Questions and Answers
Question 1: What is the rectangular equivalent of the polar point (2√2, 225°)?
- (2, −2)
- (−2, −2) (Correct answer)
- (2, 2)
- (−2, 2)
Correct answer: (−2, −2)
x = 2√2·cos 225° = 2√2·(−√2/2) = −2, y = 2√2·sin 225° = −2, giving (−2, −2).
Question 2: How many petals does the rose curve r = cos(5θ) have?
- 5 (Correct answer)
- 10
- 3
- 15
Correct answer: 5
r = cos(nθ) has n petals when n is odd.
Question 3: The polar equation r = 4/(1 + cos θ) represents which conic section?
- Ellipse
- Hyperbola
- Parabola (Correct answer)
- Circle
Correct answer: Parabola
When e = 1 in r = ed/(1 + e cos θ), the conic is a parabola.
Question 4: Evaluate: Find θ in [0, 2π) where the polar curve r = sin(2θ) passes through the pole.
- θ = 0 only
- θ = π/2 and π
- θ = 0, π/2, π, 3π/2 (Correct answer)
- θ = π/4 and 3π/4
Correct answer: θ = 0, π/2, π, 3π/2
r = 0 when sin(2θ) = 0, i.e., 2θ = 0, π, 2π, 3π → θ = 0, π/2, π, 3π/2.
Question 5: Which symmetry does r = 4 + 4 cos θ exhibit?
- Symmetric about θ = π/2
- Symmetric about the polar axis (Correct answer)
- Symmetric about the pole
- No symmetry
Correct answer: Symmetric about the polar axis
Replacing θ with −θ leaves the equation unchanged, so it is symmetric about the polar axis.
Question 6: What is the slope of the tangent line to r = 2 at θ = π/4?
- 0
- 1
- −1 (Correct answer)
- Undefined
Correct answer: −1
For a circle r = 2, the tangent at θ = π/4 is perpendicular to the radius, giving slope −cot(π/4) = −1.
Question 7: Convert the rectangular equation x² − y² = 4 to polar form.
- r² = 4 cos(2θ) (Correct answer)
- r² = 4 sin(2θ)
- r cos(2θ) = 2
- r² = 4/(cos θ − sin θ)
Correct answer: r² = 4 cos(2θ)
x² − y² = r²cos²θ − r²sin²θ = r²cos(2θ) = 4, so r² = 4/cos(2θ)... actually x²−y²=r²cos(2θ)=4.
Trigonometry Polar Coordinates 3
What is the rectangular equivalent of the polar point (2√2, 225°)?