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Find (dy)/(dx) at t=(2pi/3) when x = ...

Find `(dy)/(dx)` at `t=(2pi/3)` when `x = 10 (t – sin t) and y = 12 (1 – cos t)`.

Text Solution

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Here,

`dy/dx=(dy/dt)/(dx/dt)`


Calculating `dy/dt`


`y=12(1-cost)`

`y=12-12cost`

Differentiating w.r.t. x

`dy/dt=(d(12-12cost))/dt`

`dy/dt=(d(12))/dt-12(d(cost))/dt`

`dy/dt=0-12(-sint)`

`dy/dt=12sint`


Calculating `dx/dt`

`x=10(t-sint)`

`x=10t-10sint`

Differentiating w.r.t. x

`dx/dt=(d(10t-10sint))/dt`

`dx/dt=(d(10t))/dt-(d(10sint))/dt`

`dx/dt=10-10cost`

`dx/dt=10(1-cost)`


`dy/dx=(dy/dt)/(dx/dt)`

`dy/dx=(12sint)/(10(1-cost))`

`dy/dx=(6sint)/(5(1-cost))`

`dy/dx=(6.2sin(t/2)cos(t/2))/(5(2sin^2(t/2)))`

`dy/dx=(6cos(t/2))/(5sin(t/2))`

`dy/dx=6/5cot(t/2)`
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