Question

if x=a cos theta ,y=a sin theta then find dy/dx​

Answer 1

Solution

Given:-

$\implies\sf\: x = acos \theta$

$\sf \implies \: y = asin \theta$

To find

$\sf \implies \: \dfrac{dy}{dx}$

Now Take x Differentiate W.R.T Θ

$\sf \implies \dfrac{dx}{d \theta} = acos \theta$

$\sf \implies \: \dfrac{dacos \theta}{d \theta} \: \: \: \: \: where \: \: a \: is \: constant$

We can write as

$\sf \implies \: \dfrac{a \: dcos \theta}{d \theta}$

We get

$\sf \implies \dfrac{dx}{d \theta} = - a \: sin \theta$

Now Take y Differentiate W.R.T Θ

$\sf \implies \dfrac{dy}{d \theta} = a \: sin \theta$

$\sf \implies \: \dfrac{a \: d \: sin \theta}{d \theta}$

$\sf \implies \: \dfrac{dy}{ d\theta} = a \: cos \theta$

Now we have to find dy/dx , so we can write as

$\sf \implies \: \dfrac{ \dfrac{dy}{d \theta} }{ \dfrac{dx}{d \theta} } \implies \dfrac{dy}{d \theta} \times \dfrac{d \theta}{dx} \implies \: \dfrac{dy}{dx}$

We get

$\sf \implies \: \dfrac{dy}{dx} = \dfrac{ \dfrac{dy}{d \theta} }{ \dfrac{dx}{d \theta} } = \dfrac{acos \theta}{ - a \sin \theta} = - cot \theta$

Answer is

$\sf \implies \: - cot \theta$