Question

`\sf \large \: find \: (dy)/(dx) \: if \: x=a ( \Theta + sin \Theta ) , y= a (1- Cos \Theta ) \: at \: \Theta = (π)/(2) \\ \\ \\ \\ \\`

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(dx)/(dθ ) = a \: (d)/(dθ) (θ + \sin\theta )`

`\: \:`

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

`\: \:`

Now y = a ( 1-cosθ)

`\: \:`

Diff w.r.t " θ " we get .

`\: \:`

• 
  `\rm \large (dy)/(d \theta) = a (d)/(d \theta) (1 - cos \theta)`

`\: \: \:`

• 
  `\boxed{ \rm \large \underline{ (dy)/(d \theta) = a \sin \theta}}`

`\: \: \:`

From eqn ( 1 ) & ( 2 )

`\: \:`

• 
  `\rm \large (dy)/(dx) = ( (dy)/(d \theta) )/((dx)/(d \theta))`

`\: \: \:`

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

`\: \:`

• 
  `\rm \large \: = (sin \theta)/(1 + \cos \theta )`

`\: \: \:`

• 
  `\rm \large \: (2 \sin( (\theta )/(2) ) cos(\theta )/(2) )/(2 \: cos ^(2) (\theta )/(2)) \: \: ......(sin \: a \: = 2 \: sin (a)/(2) \: cos (a)/(2) 1 + \: cos \: a \: = 2cos ^(2) (a)/(2) )`

`\: \:`

• 
  `\rm \large \: (dy)/(dx) = (sin ( \theta)/(2) )/(cos ( \theta)/(2) )`

`\: \:`

• 
  `\rm \large \: (dy)/(dx) = tan ( \theta)/(2)`

`\: \:`

• 
  `\rm \large \: ( (dy)/(dx) ) = tan ( ( \theta)/(2) )/(2)`

`\: \:`

• 
  `\rm \large \: = tan( ( \theta)/(4) )`

`\: \:`

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

`\: \:`

Hope Helps!:)