Answer
a)
$\frac{dy/dθ}{dx/dθ} = \frac{\frac{d}{dθ}(rsin(θ))}{\frac{d}{dθ}(rcos(θ))}$=$\frac{\frac{d}{dθ}(sin(θ)+rcos(θ))}{\frac{d}{dθ}(cos(θ)-rsin(θ))}$.
b)
$A=\int_{a}^{b} \frac{1}{2}r^{2}dθ$.
c)
$L= \int_{a}^{b} \sqrt ((\frac{dx}{dθ})^{2}+(\frac{dy}{dθ})^{2})dθ$ = $\int_{a}^{b} \sqrt (r^{2}+(\frac{dr}{dθ})^{2})dθ$
Work Step by Step
a)
The slope of the tangent line to a polar curve is defined as $\frac{dy}{dx}$. The slope in terms of polar coordinates is $\frac{dy/dθ}{dx/dθ} = \frac{\frac{d}{dθ}(rsin(θ))}{\frac{d}{dθ}(rcos(θ))}$=$\frac{\frac{d}{dθ}(sin(θ)+rcos(θ))}{\frac{d}{dθ}(cos(θ)-rsin(θ))}$.
b)
The area of a region bounded by a polar curve is: $A=\int_{a}^{b} \frac{1}{2}r^{2}dθ$.
c)
The length of a polar curve is: $L= \int_{a}^{b} \sqrt ((\frac{dx}{dθ})^{2}+(\frac{dy}{dθ})^{2})dθ$ = $\int_{a}^{b} \sqrt (r^{2}+(\frac{dr}{dθ})^{2})dθ$
