I'm struggling with this problem, I'm still only on part (a). I tried X=rcos(theta) Y=rsin(theta) but I don't think I'm doing it right.
Curve C has polar equation r=sin(${\theta}$)+cos(${\theta}$).
(a) Write parametric equations for the curve C.
$\left\{\begin{matrix}
x=
\\ y=
\end{matrix}\right.$(b) Find the slope of the tangent line to C at its point where
${\theta}$ = $\frac{\pi}{2}$.(c) Calculate the length of the arc for 0 $\leq {\theta} \leq {\pi}$ of that
same curve C with polar equation r=sin(${\theta}$)+cos(${\theta}$).
Best Answer
You can rewrite $x=r \cos \theta$ as $r=\frac{x}{\cos\theta}$ and plug that in. You immediately get $$x=\sin\theta\cos\theta+\cos^2\theta$$ Doing the same trick for $r=\frac{y}{\sin\theta}$ gives you $$y=\sin^2\theta+\sin\theta\cos\theta$$
From here on it's not hard - the slope of the tangent is $\frac{dy/d\theta}{dx/d\theta}$