Trigonometry Polar Coordinates 4

Question 1

Find the area of one petal of r = sin(3θ).

Accepted Answer

π/12

Answer

Area = ½∫₀^(π/3) sin²(3θ) dθ = ½·(π/6)·(1) = π/12.

Question 2

The inner loop of the limaçon r = 1 − 2 cos θ exists when r < 0. What is the angular range of the inner loop?

Accepted Answer

π/3 < θ < 5π/3

Answer

r < 0 when cos θ > 1/2, i.e., θ ∈ (−π/3, π/3)... the inner loop maps to where cos θ > 1/2: θ ∈ (5π/3, 2π)∪(0, π/3) and equivalently traced over π/3 < θ < 5π/3 for r < 0.

Question 3

Which polar equation produces a lemniscate?

Accepted Answer

r² = 4 cos(2θ)

Answer

r² = a² cos(2θ) is the standard form of a lemniscate.

Question 4

At what values of θ ∈ [0, 2π) does r = 2 + 2 sin θ reach its maximum r?

Accepted Answer

θ = π/2

Answer

r is maximized when sin θ = 1, i.e., θ = π/2, giving r = 4.

Question 5

What is the polar equation of the vertical line x = 3?

Accepted Answer

r cos θ = 3

Answer

Since x = r cos θ, setting x = 3 gives r cos θ = 3.

Trigonometry Practice Test

Trigonometry Practice Test

Trigonometry Polar Coordinates 4