Horizontal and vertical tangents to a parametric curve

Question

I'm not sure what my procedure should be when solving this problem:

1. find all points with a horizontal tangent
2. find all points with a vertical tangent
3. find all inflection points

$$x(t) = \sin(t),\quad y(t)= t $$

When trying to solve these kinds of problems what steps should I take? I know that $x'(t) = \cos(t) $ and $y'(t) = 1$.

Answer 1

By chain rule you have that $$ \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}. $$ So $$ \frac{dy}{dx}=\frac{1}{\cos t}. $$ Can you take it from there?