Question

How to calculate the slope of the line tangent to the graph of?

r = 2+4sinθ at the point corresponding to θ=pi/6

Answer 1

`3 sqrt(3)`

Explanation:

The slope of the tangent line to any curve is `dy/dx`

Since we know how to convert between Cartesian and polar coordinates, we can do a chain rule substitution:
`dy/dx = (dy/(d theta))/(dx/(d theta)) = (d/(d theta) (r sin theta))/(d/(d theta) (r cos theta)) = ((dr)/(d theta) sin theta + r costheta)/((dr)/(d theta) costheta - r sin theta)`

This formula works in general, but we can apply it here using the formulae provided:
`theta = pi/6 implies cos(theta) = sqrt(3)/2 and sin(theta) = 1/2`
`r(theta) = 2 + 4 * 1/2 = 4 and (dr)/(d theta) = 4costheta = 2sqrt(3)`
Therefore,
`dy/dx = (2 sqrt(3) * 1/2 + 4 * sqrt(3)/2)/(2 sqrt(3) * sqrt(3)/2 - 4 * 1/2) = (sqrt(3) + 2sqrt(3))/(3-2) = 3 sqrt(3)`