Final answer:
The curvature of the curve r = 4sin(3t) is κ = 3|cos(3t)|.
Step-by-step explanation:
The curvature of a curve is defined as the rate of change in the direction of the tangent vector with respect to arc length. To find the curvature of the curve r = 4sin(3t), we need to find the derivative of the vector r with respect to t, and then divide it by the magnitude of the derivative.
Given r = 4sin(3t), the derivative of r with respect to t is dr/dt = 12cos(3t). The magnitude of the derivative is |dr/dt| = √(12^2cos^2(3t)) = √(144cos^2(3t)) = 12|cos(3t)|.
Therefore, the curvature of the curve r = 4sin(3t) is given by κ = |dr/dt|/|r| = 12|cos(3t)|/4 = 3|cos(3t)|.