Question

%20if%20%5C%3A%20%20x%3Da%20%28%20%20%5CTheta%20%2B%20sin%20%20%5CTheta%20%29%20%2C%20y%3D%20a%20%281-%20Cos%20%20%5CTheta%20%29%20%5C%3A%20%20at%20%5C%3A%20%20%20%5CTheta%20%3D%20%5Cfrac%7B%CF%80%7D%7B2%7D%20%20%5C%5C%20%20%5C%5C%20%20%5C%5C%20%20%5C%5C%20%20%5C%5C%20" id="TexFormula1" title=" \sf \large \: find \: \frac{dy}{dx} \: if \: x=a ( \Theta + sin \Theta ) , y= a (1- Cos \Theta ) \: at \: \Theta = \frac{π}{2} \\ \\ \\ \\ \\ " alt=" \sf \large \: find \: \frac{dy}{dx} \: if \: x=a ( \Theta + sin \Theta ) , y= a (1- Cos \Theta ) \: at \: \Theta = \frac{π}{2} \\ \\ \\ \\ \\ " align="absmiddle" class="latex-formula">
Kindly Don't Spàm!
Thanks!!!!​

Answer 1

Given :

$\: \: \:$

• $\rm \large \: x = a ( \Theta + Sin \Theta )$

$\: \: \:$

• $\rm \large y = a ( 1 - cos \: \Theta )$

$\: \:$

Now , x = a ( θ + sin θ )

$\: \:$

Diff w.r.t " θ "

$\: \: \:$

• $\rm \large\frac{dx}{dθ } = a \: \frac{d}{dθ} (θ + \sin\theta )$

$\: \:$

• $\boxed{ \rm \large\underline{ \frac{dx}{d \theta} = a(1 + \cos \theta ) }}$

$\: \:$

Now y = a ( 1-cosθ)

$\: \:$

Diff w.r.t " θ " we get .

$\: \:$

• $\rm \large \frac{dy}{d \theta} = a \frac{d}{d \theta} (1 - cos \theta)$

$\: \: \:$

• $\boxed{ \rm \large \underline{ \frac{dy}{d \theta} = a \sin \theta}}$

$\: \: \:$

From eqn ( 1 ) & ( 2 )

$\: \:$

• $\rm \large \frac{dy}{dx} = \frac{ \frac{dy}{d \theta} }{\frac{dx}{d \theta}}$

$\: \: \:$

• $\: \: \: \rm \large = \frac{ \cancel{a} \: sin \theta}{ \cancel a \: (1 + cos \theta)}$

$\: \:$

• $\rm \large \: = \frac{sin \theta}{1 + \cos \theta }$

$\: \: \:$

• $\rm \large \: \frac{2 \sin( \frac{\theta }{2} ) cos\frac{\theta }{2} }{2 \: cos ^{2} \frac{\theta }{2}} \: \: ......(sin \: a \: = 2 \: sin \frac{a}{2} \: cos \frac{a}{2} 1 + \: cos \: a \: = 2cos ^{2} \frac{a}{2} )$

$\: \:$

• $\rm \large \: \frac{dy}{dx} = \frac{sin \frac{ \theta}{2} }{cos \frac{ \theta}{2} }$

$\: \:$

• $\rm \large \: \frac{dy}{dx} = tan \frac{ \theta}{2}$

$\: \:$

• $\rm \large \: ( \frac{dy}{dx} ) = tan \frac{ \frac{ \theta}{2} }{2}$

$\: \:$

• $\rm \large \: = tan( \frac{ \theta}{4} )$

$\: \:$

• $\boxed{ \rm \large \underline{ ( \frac{dy}{dx} ) = \frac{\pi}{2} = 1}}$

$\: \:$

Hope Helps!:)