FMAT pure1 polar coordinates

Question 1

What are polar coordinates?

Answer 1

Polar coordinates describe a point using (r, θ), where:
- r is the distance from the pole (origin).
- θ is the angle from the initial line (measured anticlockwise in radians)

Question 2

How do you convert polar coordinates to Cartesian coordinates?

Answer 2

x = r * cos(θ)
y = r * sin(θ)

Question 3

How do you convert Cartesian coordinates to polar coordinates?

Answer 3

r = √(x² + y²)
θ = tan⁻¹(y / x)
Ensure θ is in the correct quadrant.

Question 4

Example: Convert Cartesian coordinates (3, 4) to polar coordinates.

Answer 4

r = √(3² + 4²) = 5
θ = tan⁻¹(4 / 3) ≈ 0.93 radians
Polar coordinates: (5, 0.93)

Question 5

Example: Convert polar coordinates (4, π/6) to Cartesian coordinates.

Answer 5

x = 4 * cos(π/6) = 2√3
y = 4 * sin(π/6) = 2
Cartesian coordinates: (2√3, 2)

Question 6

What is the general approach to sketching polar curves?

Answer 6

1. Identify key values of θ where r is zero, maximum, or minimum.
2. Use symmetry and the behavior of trigonometric functions.
3. Sketch the curve, showing critical points and shape.

Question 7

Example: Sketch the polar curve r = 2 + 4sin(θ).

Answer 7

• Maximum r: r = 6 when sin(θ) = 1.
• Minimum r: r = 2 when sin(θ) = 0.
• check desmos

Question 8

How do you find the Cartesian equation of a polar curve?

Answer 8

Use:
x = r * cos(θ), y = r * sin(θ), and r² = x² + y²
Substitute and simplify as needed.

Question 9

Example: Convert the polar curve r = 2 + 2cos(θ) to Cartesian form.

Answer 9

Multiply through by r: r² = 2r + 2r * cos(θ)
Substitute r² = x² + y² and r * cos(θ) = x:
x² + y² = 2√(x² + y²) + 2x

Question 10

Example: Find the polar equation of a circle with centre (2, 0) and radius 2.

Answer 10

Using symmetry, the polar equation is:
r = 4cos(θ)

Question 11

What is the formula for the area of a sector in polar coordinates?

Answer 11

Area = ∫(1/2) r^2 dθ
The limits of integration are values of θ.

Question 12

Why is the double-angle identity often used in polar area calculations?

Answer 12

To simplify trigonometric expressions like sin^2(θ) or cos^2(θ) before integration:
sin^2(θ) = (1 - cos(2θ)) / 2
cos^2(θ) = (1 + cos(2θ)) / 2

Question 13

Find the area enclosed by one loop of the curve r = sin(2θ).

Answer 13

Area = (1/2) ∫ [sin^2(2θ)] dθ from 0 to π/2
Use sin^2(2θ) = (1 - cos(4θ)) / 2:
Area = (1/2) ∫ [(1 - cos(4θ))/2] dθ = π / 8

Question 14

How is the gradient of a polar curve found?

Answer 14

Convert to Cartesian coordinates and use dy/dx. In polar form:
x = r * cos(θ), y = r * sin(θ)
dy/dx = (dy/dθ) / (dx/dθ)

Question 15

Example: Find the gradient of r = 2sin(2θ) at θ = π/4.

Answer 15

1. x = r * cos(θ) = 2sin(2θ) * cos(θ)
2. y = r * sin(θ) = 2sin(2θ) * sin(θ)
3. Gradient (dy/dx) at θ = π/4 is -1.

Question 16

How do you find tangents parallel or perpendicular to the initial line?

Answer 16

• Tangents parallel to the initial line: dy/dθ = 0.
• Tangents perpendicular to the initial line: dx/dθ = 0.

Question 17

Example: For r = sin(2θ), find the equations of tangents parallel and perpendicular to the initial line.

Answer 17

1. dx/dθ = 0 gives x = ±4/(3√3) (perpendicular).
2. dy/dθ = 0 gives y = ±4/(3√3) (parallel).

Question 18

What is the process to find the area of a polar region?

Answer 18

1. Identify the limits of θ for the region.
2. Use Area = ∫ (1/2) r^2 dθ.
3. Simplify using trigonometric identities if needed.

Question 19

Why is implicit differentiation often needed in polar coordinates?

Answer 19

Polar equations convert to Cartesian forms that involve both x and y. Implicit differentiation is used to find dy/dx.

Question 20

Example: Convert r = 2sin(2θ) to Cartesian form.

Answer 20

Multiply by r: r^2 = 2r * sin(2θ)
Using x = r * cos(θ) and y = r * sin(θ):
x^2 + y^2 = 2xy