Question

Sketch the curve and find all points on the cardioid
r=a(1+cos(theta)) where the tangent line is horizontal.

Answer 1

r = a(1 + cos θ)
If the tandent line is horizontal, then dr/dθ = 0
dr/dθ = -a sin θ = 0
sin θ = 0
θ = 0, π

For θ = 0: r = a(1 + cos 0) = a(1 + 1) = 2a
For θ = π: r = a(1 + cos π) = a(1 - 1) = 0

Therefore, the points where the tangent line is horizontal are (2a, 0) and (0, π)