Given x = cost y=sin2t, how do you find the dy/dx terms parameter t and find the values parameter t points dy/dx = 0?

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Given x = cost y=sin2t, how do you find the dy/dx terms parameter t and find the values parameter t points dy/dx = 0?

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Answer 1 · 2016-07-03T06:32:14

$\frac{d y}{d x} = 0$ $(dy)/(dx)=0$ at $t = (2 m + 1) \frac{π}{4}$ $t=(2m+1)pi/4$

Explanation:

In parametric equations $x = x (t)$ $x=x(t)$ and $y = y (t)$ $y=y(t)$ , $\frac{d y}{d x} = \frac{\frac{d y}{d t}}{\frac{d x}{d t}}$ $(dy)/(dx)=((dy)/(dt))/((dx)/(dt))$

As $y = sin 2 t$ $y=sin2t$ , $\frac{d y}{d t} = cos 2 t \times 2 = 2 cos 2 t$ $(dy)/(dt)=cos2txx2=2cos2t$

and as $x = cos t$ $x=cost$ , $\frac{d x}{d t} = - sin t$ $(dx)/(dt)=-sint$

Hence $\frac{d y}{d x} = \frac{- 2 cos 2 t}{sin t}$ $(dy)/(dx)=(-2cos2t)/sint$

As $sin t \neq 0$ $sint!=0$ , when $t = n π$ $t=npi$

$\frac{d y}{d x} = 0$ $(dy)/(dx)=0$ , when $cos 2 t = 0$ $cos2t=0$ but $t \neq n π$ $t!=npi$ i.e. $2 t = (2 m + 1) \frac{π}{2}$ $2t=(2m+1)pi/2$

or $t = (2 m + 1) \frac{π}{4}$ $t=(2m+1)pi/4$ , where $m$ $m$ is an integer

But note that $x$ $x$ and $y$ $y$ both are sinusoidal functions and hence their domain is limited to $[- 1, 1]$ $[-1,1]$ and hence as $x = cos t$ $x=cost$ , $\frac{d y}{d x} = 0$ $(dy)/(dx)=0$ at $x = \pm \frac{1}{\sqrt{2}}$ $x=+-1/sqrt2$

graph{2xsqrt(1-x^2) [-2.527, 2.473, -1.11, 1.39]}

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Given x = cost y=sin2t, how do you find the dy/dx terms parameter t and find the values parameter t points dy/dx = 0?

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