Question

Consider the set of parametric equations x=sin(t), y=sin(2t). What is the equation of the tangent line at the origin with positive slope?

Answer 1

`y=2x`

Explanation:

Slope of tangent at a point `(x_0,f(x_0))` on the curve `y=f(x)` is given by the value of the derivative `(df)/(dx)` at `(x_0,f(x_0))`.

Here we are given a parametric equation of the type `(x(t),y(t))`. Slope of such parametric equation is given by `((dy)/(dt))/((dx)/(dt))`

As `x=sint` and `y=sin2t`, `(dy)/(dt)=2cos2t` and `(dx)/(dt)=cost`

Now at origin i.e. `(0,0)`, we have `t=0`, as it makes both `x` and `y` equal to `0`, and hence slope of tangent is given by

`(dy)/(dx)=((dy)/(dt))/((dx)/(dt))=(2cos2t)/cost`

and hence value of slope at `t=0` is `2` (observe that slope is positive) and equation of tangent is `y=2x`

graph{(y-2xsqrt(1-x^2))(y-2x)=0 [-2.5, 2.5, -1.25, 1.25]}