Final answer:
To find the slope of the tangent line to the polar curve r(θ) = 3csc(2θ) at θ = π/3, calculate the derivatives dr/dθ and r dθ/dθ at θ = π/3 and then use the formula slope = (dr/dθ)/(r dθ/dθ). If the slope is undefined, state that the tangent line is vertical.
Step-by-step explanation:
To find the slope of the tangent line for the polar curve r(θ) = 3csc(2θ) at the point θ = π/3, you first need to understand that the slope of a curve at a point is equal to the slope of the straight line tangent to the curve at that point. In the context of polar coordinates, this requires the use of calculus, specifically the derivative of the polar curve with respect to θ.
Slope of the Tangent Line,
- Derive dr/dθ and r dθ/dθ which give the derivatives of r with respect to θ.
- Evaluate these derivatives at the point θ = π/3.
- Use the polar slope formula: slope = (dr/dθ)/(r dθ/dθ) at θ = π/3 to find the slope.
If the tangent line is determined to be vertical, you would state that the slope is "vertical" which implies it is undefined since vertical lines do not have a numerical slope. However, in this case, we must complete the calculation to determine the nature of the slope.