Determine the equation of the tangent line to r = 3 + 8 sinθ at θ = Π/6.

Solution

We'll first need the subsequent derivative.

dr/dθ = 8 cosθ

The formula for the derivative dy/dx becomes,

dy/dx = 8cosθ sinθ + (3+8 sinθ) cos θ / 8 cos² θ - (3+ 8 sinθ) sinθ

= 16 cosθ sinθ + 3cosθ / 8 cos²θ - 3sinθ - 8 sin² θ

The slope of the tangent line is as follow:

Here now, at  θ = Π/6 we have r = 7 .  We'll need to obtain the corresponding x-y coordinates thus we can get the tangent line.

 x = 7 cos (Π/6)

= 7√3 / 2

y = 7sin (Π/6)

= 7/2

y = 7/2 + 11√3 / 5 (x - 7√3 /2)

On behalf of completeness here is a graph of the curve and the tangent line.