Consider the graphs of r = cos(3θ) + 2 and r = 3. Let A = (3, 0) and B = (1/2, sqrt(3)/2)

Question

1.  Find the equation of the tangent line to the graph at point B and explain what that tells you about the movement of the particle.
2. Find the length (arclength) of the above trip from A to B on the graph of r = cos(3θ) + 2 using polar coordinates. Setup but don’t solve the definite integral.
3.  Find the length (arclength) of the above trip from A to B on the graph of r = cos(3θ) + 2 using parametric coordinates. Setup but don’t solve the definite integral.

Answer 1

We have two curves r = cos(3θ) +2 and r=3. r = 3 is a circle with radius 3. Hence The point B with cartesian coordinates (1/2, √3/2) would lie on the curve cos (3θ ) +2. The polar coordinates for this point are (1, pi/3).

1. The derivate of r with respect to θ is -3sin(3θ). At θ = pi/3 this is 0. The slope of the tangent line would be 0 and hence the tangent is horizontal. The equation to the tangent line would be y = √3/2.
2. The length of the curve can be found by using the formula ∫ √r² + (dr/dθ)² dθ between limits 0 to pi/3. Using a TI-84 calculator or desmos , this is equal to 3.025.
3. You may find the parameters x = r cos(θ) = cos(θ) (cos(3θ) +2) and y = r sin(θ) = sin(θ) (cos(3θ) + 2). Then use the formula ∫√(dx/dθ)² + (dy/dθ)² dθ for limits 0 to pi/3. This should give us the same answer 3.025.