Sketch a graph of the polar equation and find the tangent lines at the pole for 0 ≤ 𝜃 < 𝜋 (if any). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)r = 3 cos(3𝜃)

To sketch the graph of the polar equation r = 3cos(3θ), we need to understand that this is a rose curve with 3 petals. The number of petals in a rose curve is determined by the coefficient of θ in the cosine function. If this number is odd, the number of petals is the same as this number. If it is even, the number of petals is twice this number.

Here are the steps to sketch the graph:

1. Identify the maximum value of r. This occurs when cos(3θ) = 1, so r = 3. This is the length of each petal.

2. Identify the values of θ that make r = 0. This occurs when cos(3θ) = 0, so 3θ = π/2, 3π/2, 5π/2, etc. Solving for θ gives θ = π/6, π/2, 5π/6, etc. These are the directions of the petals.

3. Plot the points (r, θ) for these values of θ and connect them with smooth curves to form the petals.

To find the tangent lines at the pole, we need to find the values of θ for which r = 0. As mentioned above, this occurs when θ = π/6, π/2, 5π/6, etc. The tangent line at the pole in the direction of θ is the line θ = constant. So the tangent lines at the pole are θ = π/6, θ = π/2, θ = 5π/6, etc.

However, since the question asks for the tangent lines for 0 ≤ θ < π, we only consider the first two of these lines. So the tangent lines at the pole for 0 ≤ θ < π are θ = π/6 and θ = π/2.