Question

dy/dx of X=a(t+sint) y=a(1-cost)

Answer 1

Answer:

$\sf \dfrac{dy}{dx} =\dfrac{sin\:t}{(1+cos\:t)}$

Step-by-step explanation:

Given:

• x = a (t + sin t)
• y = a (1 - cos t)

To Find:

• dy/dx

Solution:

By given,

y = a (t + sin t)

Differentiating on both sides with respect to t we get,

$\sf \dfrac{dy}{dt} =\dfrac{dt}{dt} (a\:(1-cos\:t))$

$\sf \implies a\:\dfrac{dt}{dt} (1-cos\:t)$

$\implies \sf a\times -(-sin\:t)$

$\implies \sf a\:sint---(1)$

Now also by given,

x = a ( t + sin t)

Differentiate on both sides with respect to x,

$\sf \dfrac{dx}{dt} =\dfrac{d}{dt} (a\:(t+sin\:t))$

$\sf \implies a\:\dfrac{d}{dt} (t+sin\:t)$

$\sf \implies a\:(1+cos\:t)---(2)$

Now dividing equation 1 by equation 2 we get,

$\sf \dfrac{dy/dt}{dx/dt} =\dfrac{a\:sin\:t}{a\:(1+cos\:t)}$

$\sf \dfrac{dy}{dx} =\dfrac{sin\:t}{(1+cos\:t)}$

This is the required solution.

Answer 2

Answer:

cot⁻¹ [ a ( 1 + cos t ) / ( sin t ) ]

Step-by-step explanation:

Given :

x = a ( t + sin t )

= > x = a t + a sin t

Diff. w.r.t. t :

d x / d t = a ( t )' + a ( sin t )'

= > d x / d t = a + a cos t

= > d x / d t = a ( 1 + cos t )

Also given :

y = ( 1 - cos t )

Diff. w.r.t. t :

= > d y / d t = ( 1 )' - ( cos t )'

= > d y / d t = - ( - sin t )

= > d y / d t = sin t

Now as we do in parametric function :

d x / d y = ( d x / d t ) / ( d y / d t )

= > d x / d y = a ( 1 + cos t ) / ( sin t )

Given d x / d y = cot P

cot P = a ( 1 + cos t ) / ( sin t )

P = cot⁻¹ [ a ( 1 + cos t ) / ( sin t ) ]

Hence we get required answer.

Answer 3

Answer:

cot⁻¹ [ a ( 1 + cos t ) / ( sin t ) ]

Step-by-step explanation:

Given :

x = a ( t + sin t )

= > x = a t + a sin t

Diff. w.r.t. t :

d x / d t = a ( t )' + a ( sin t )'

= > d x / d t = a + a cos t

= > d x / d t = a ( 1 + cos t )

Also given :

y = ( 1 - cos t )

Diff. w.r.t. t :

= > d y / d t = ( 1 )' - ( cos t )'

= > d y / d t = - ( - sin t )

= > d y / d t = sin t

Now as we do in parametric function :

d x / d y = ( d x / d t ) / ( d y / d t )

= > d x / d y = a ( 1 + cos t ) / ( sin t )

Given d x / d y = cot P

cot P = a ( 1 + cos t ) / ( sin t )

P = cot⁻¹ [ a ( 1 + cos t ) / ( sin t ) ]

Hence we get required answer.