Question:
If \(x\) and \(y\) are connected parametrically by the equation,without eliminating the parameter,find \(\frac{dy}{dx}\).
\(x=sin\,t,y=cos\,2t\)
If \(x\) and \(y\) are connected parametrically by the equation,without eliminating the parameter,find \(\frac{dy}{dx}\).
\(x=sin\,t,y=cos\,2t\)
\(x=sin\,t,y=cos\,2t\)
Updated On: Sep 12, 2023
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Solution and Explanation
The correct answer is \(-4sin\,t\)
The given equations are \(x=sin\,t,y=cos\,2t\)
Then,\(\frac{dx}{dt}=\frac{d}{dt}(sin\,t)=cos\,t\)
\(\frac{dy}{dt}=\frac{d}{dt}(cos\,2t)=-sin\,2t.\frac{d}{dt}(2t)=-2sin\,2t\)
\(∴\frac{dy}{dx}=\frac{(\frac{dy}{dt})}{(\frac{dx}{dt})}=\frac{2sin\,2t}{cos\,t}=\frac{-2.2sin\,tcos\,t}{cos\,t}=-4sin\,t\)
The given equations are \(x=sin\,t,y=cos\,2t\)
Then,\(\frac{dx}{dt}=\frac{d}{dt}(sin\,t)=cos\,t\)
\(\frac{dy}{dt}=\frac{d}{dt}(cos\,2t)=-sin\,2t.\frac{d}{dt}(2t)=-2sin\,2t\)
\(∴\frac{dy}{dx}=\frac{(\frac{dy}{dt})}{(\frac{dx}{dt})}=\frac{2sin\,2t}{cos\,t}=\frac{-2.2sin\,tcos\,t}{cos\,t}=-4sin\,t\)
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