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