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vlabodo [156]
3 years ago
12

Find dy/dx x=a(cost +sint) , y=a(sint-cost)​

Mathematics
1 answer:
MissTica3 years ago
4 0

Answer:

\begin{aligned} \frac{dy}{dx} &= \frac{\cos(t) + \sin(t)}{\cos(t) - \sin(t)} \end{aligned} given that a \ne 0 and that \cos(t) - \sin(t) \ne 0.

Step-by-step explanation:

The relation between the y and the x in this question is given by parametric equations (with t as the parameter.)

Make use of the fact that:

\begin{aligned} \frac{dy}{dx} = \quad \text{$\frac{dy/dt}{dx/dt}$ given that $\frac{dx}{dt} \ne 0$} \end{aligned}.

Find \begin{aligned} \frac{dx}{dt} \end{aligned} and \begin{aligned} \frac{dy}{dt} \end{aligned} as follows:

\begin{aligned} \frac{dx}{dt} &= \frac{d}{dt} [a\, (\cos(t) + \sin(t))] \\ &= a\, (-\sin(t) + \cos(t)) \\ &= a\, (\cos(t) - \sin(t))\end{aligned}.

\begin{aligned} \frac{dx}{dt} \ne 0 \end{aligned} as long as a \ne 0 and \cos(t) - \sin(t) \ne 0.

\begin{aligned} \frac{dy}{dt} &= \frac{d}{dt} [a\, (\sin(t) - \cos(t))] \\ &= a\, (\cos(t) - (-\sin(t))) \\ &= a\, (\cos(t) + \sin(t))\end{aligned}.

Calculate \begin{aligned} \frac{dy}{dx} \end{aligned} using the fact that \begin{aligned} \frac{dy}{dx} = \text{$\frac{dy/dt}{dx/dt}$ given that $\frac{dx}{dt} \ne 0$} \end{aligned}. Assume that a \ne 0 and \cos(t) - \sin(t) \ne 0:

\begin{aligned} \frac{dy}{dx} &= \frac{dy/dt}{dx/dt} \\ &= \frac{a\, (\cos(t) + \sin(t))}{a\, (\cos(t) - \sin(t))} \\ &= \frac{\cos(t) + \sin(t)}{\cos(t) - \sin(t)}\end{aligned}.

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