Curve in parametric form with Polar Coordinates

Example: Find the points on the curve r = 3 cos θ where the tangent line is

horizontal or vertical.

Strategy: First determine x and y in terms of r and θ in order to find dy/dx

Recall x = r cos θ and y = r sin θ where r = 3 cos θ

So x = 3 cos² θ y = 3 sin θ cos θ Note: Here the parameter is θ.

Strategy: Calculate the derivative dy/dx = dy/dθ / dx/dθ provided that dx/dθ ≠ 0

dy/dθ = 3 cos² θ - 3 sin² θ = 3 (cos² θ - sin² θ ) = 3 cos 2θ

dx/dθ = - 6 cos θ sin θ = -3 sin 2θ So dy/dx = 3 cos 2θ / -3 sin 2θ = - cot 2θ

Now the tangent line is horizontal whenever dy/dx = 0. Implies dy/dθ = 0

Also the tangent line is vertical whenever dy/dx → ± ∞. Implies dx/dθ = 0

See the curve r = 3 cos θ (circle) shown below (along with dotted tangent lines)

Result for horizontal tangents: (At A and B)

So for dy/dθ = 0 = 3 cos 2θ yields 2θ = π/2 and 3π/2 at A and at B

(r,θ) = (3/√2,π/4) and (-3/√2,3π/4)

Result for vertical tangents: (At C and D)

So for dx/dθ = 0 = -3 sin 2θ yields 2θ = 0 and π at C and D

(r,θ) = (0,π/2) and (3,0)

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