Trigonometry Polar Coordinates 2 — Questions and Answers
Question 1: What is the polar form of the point (−3, 0) in rectangular coordinates?
- (3, 0°)
- (3, 90°)
- (3, 180°) (Correct answer)
- (3, 270°)
Correct answer: (3, 180°)
The point (−3, 0) lies on the negative x-axis, so r = 3 and θ = 180°.
Question 2: Which polar equation represents a circle of radius 5 centered at the origin?
- θ = 5
- r = 5 (Correct answer)
- r = 5θ
- r² = 5
Correct answer: r = 5
r = 5 is a circle centered at the origin with radius 5 in polar coordinates.
Question 3: Convert r = 6 sin θ to rectangular form.
- x² + (y − 3)² = 9 (Correct answer)
- x² + y² = 6y
- (x − 3)² + y² = 9
- x² + y² = 36
Correct answer: x² + (y − 3)² = 9
Multiply both sides by r: r² = 6r sin θ → x² + y² = 6y → x² + (y−3)² = 9.
Question 4: What is the distance between polar points (4, 30°) and (4, 150°)?
- 4
- 4√2
- 4√3 (Correct answer)
- 8
Correct answer: 4√3
Using the polar distance formula: d² = 16 + 16 − 2(16)cos(120°) = 48, so d = 4√3.
Question 5: The polar curve r = 1 + sin θ is called a:
- Lemniscate
- Rose curve
- Cardioid (Correct answer)
- Limaçon with inner loop
Correct answer: Cardioid
r = 1 + sin θ is a cardioid, a special limaçon where the added constant equals the coefficient of sin θ.
Question 6: Which of the following is an equivalent representation of the polar point (5, π/3)?
- (−5, 4π/3)
- (5, 7π/3)
- (−5, π/3)
- Both A and B (Correct answer)
Correct answer: Both A and B
Adding 2π gives (5, 7π/3); adding π and negating r gives (−5, 4π/3); both are valid.
Question 7: What is the area enclosed by r = 3 cos θ?
- 3π/2
- 9π/4 (Correct answer)
- 9π/2
- 3π
Correct answer: 9π/4
r = 3 cos θ is a circle of radius 3/2, so area = π(3/2)² = 9π/4.
Trigonometry Polar Coordinates 2
What is the polar form of the point (−3, 0) in rectangular coordinates?