Which of the following are the equations of the tangent lines to the polar curve r = 2cos(3θ) at the pole?
Correct: A
Tangent lines at the pole occur when r = 0. We need to find the values of θ for which r = 0.
Set r = 0:
2cos(3θ) = 0
cos(3θ) = 0
The general solutions for cos(x) = 0 are x = π/2 + nπ, where n is an integer.
So, 3θ = π/2, 3π/2, 5π/2, 7π/2, ...
Divide by 3 to find θ:
θ = π/6, 3π/6, 5π/6, 7π/6, ...
Simplify:
θ = π/6, π/2, 5π/6, 7π/6, ...
For the rose curve r = 2cos(3θ), there are 3 petals. The tangent lines at the pole correspond to the angles at which the petals meet the pole. We typically consider values of θ in the interval [0, π) or [0, 2π) to identify distinct tangent lines.
Distinct tangent lines for 0 ≤ θ < π are θ = π/6, θ = π/2, θ = 5π/6.
If we go beyond π, θ = 7π/6 would be the same line as θ = π/6 because 7π/6 = π/6 + π.
The final answer is A.