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?

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Answer 1 · 2017-11-11T13:07:33

$y=2x$

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]}