Final answer:
The slope of the tangent line to the polar curve r = 6 sin θ at θ = π/6 is found through conversion to Cartesian coordinates and differentiation, resulting in a slope of 3√3.
Step-by-step explanation:
To find the slope of the tangent line to the polar curve r = 6 sin θ at the point specified by θ = π/6, we can use the formulas x = r × cos(θ) and y = r × sin(θ) to convert the polar equation into Cartesian coordinates. Once converted, we differentiate to get the slope at the specific point. To do this, we first find the derivative of r with respect to θ, which is dr/dθ, and then use the formula for the slope of the tangent line in polar coordinates, which is (dr/dθ)sin(θ) + rcos(θ). At θ = π/6, r is 6 sin(π/6) = 6(1/2) = 3, and dr/dθ is 6 cos(π/6) = 6(√3/2) = 3√3. Therefore, the slope at θ = π/6 is (3√3)(1/2) + (3)(√3/2), which simplifies to 3√3.