Question
If x and y are connected parametrically by the equations given in Exercise without eliminating the parameter, Find $\frac{\text{dy}}{\text{dx}}.$
$\text{x}=\sin\text{t, y}=\cos2\text{t}$
$\text{x}=\sin\text{t, y}=\cos2\text{t}$
✓
Answer
The given equations are $\text{x}=\sin\text{ t and y}=\cos=2\text{t}$
Then, $\frac{\text{dx}}{\text{dt}}=\frac{\text{d}}{\text{dt}}(\sin\text{t)}=\cos\text{t}$
$\frac{\text{dy}}{\text{dt}}=\frac{\text{d}}{\text{dt}}(\cos2\text{t})=-\sin2\text{t}.\frac{\text{d}}{\text{dt}}(2\text{t})=-2\sin2\text{t}$
$\therefore\ \frac{\text{dy}}{\text{dx}}=\frac{\Big(\frac{\text{dy}}{\text{dt}}\Big)}{\Big(\frac{\text{dx}}{\text{dt}}\Big)}=\frac{-2\sin2\text{t}}{\cos\text{t}}=\frac{-2.2\sin\text{t}\cos\text{t}}{\cos\text{t}}=-4\sin\text{t}$
Then, $\frac{\text{dx}}{\text{dt}}=\frac{\text{d}}{\text{dt}}(\sin\text{t)}=\cos\text{t}$
$\frac{\text{dy}}{\text{dt}}=\frac{\text{d}}{\text{dt}}(\cos2\text{t})=-\sin2\text{t}.\frac{\text{d}}{\text{dt}}(2\text{t})=-2\sin2\text{t}$
$\therefore\ \frac{\text{dy}}{\text{dx}}=\frac{\Big(\frac{\text{dy}}{\text{dt}}\Big)}{\Big(\frac{\text{dx}}{\text{dt}}\Big)}=\frac{-2\sin2\text{t}}{\cos\text{t}}=\frac{-2.2\sin\text{t}\cos\text{t}}{\cos\text{t}}=-4\sin\text{t}$
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