if x=2 cos t - cos 2t, y=2 sin t – sin 2t find d2y/dx2 at t=pi/2

Question

Rinkoo Gupta · Answer

dx/dt=-2sint+2sin2t

dy/dt=2cost-2cos2t

dy/dx=(dy/dt)/(dx/dt)=(cost-cos2t)/(-sint+sin2t)

dy/dx=2sin3t/2.sint/2/2cos3t/2sint/2

=tan3t/2

d²y/dx²=d/dx(dy/dx)

=d/dt(dy/dx).dt/dx

=(3/2).sec²3t/2.1/(-2sint+2sin2t)

=3/4.1/cos²(3t/2).1/(sin2t-sint)

=3/4.1/cos²(3t/2).1/2cos3t/2.sint/2

=(3/8).1/(cos³3t/2.sint/2

put t= pi/2

d²y/dx²=3/8. 1/(cos³(3pi/4).sin(pi/4))

=-3/2

Thanks & Regards

Rinkoo Gupta

AskIITians Faculty

ankit singh · Answer

dx/dt=-2sint+2sin2t

dy/dt=2cost-2cos2t

dy/dx=(dy/dt)/(dx/dt)=(cost-cos2t)/(-sint+sin2t)

dy/dx=2sin3t/2.sint/2/2cos3t/2sint/2

=tan3t/2

d²y/dx²=d/dx(dy/dx)

=d/dt(dy/dx).dt/dx

=(3/2).sec²3t/2.1/(-2sint+2sin2t)

=3/4.1/cos²(3t/2).1/(sin2t-sint)

=3/4.1/cos²(3t/2).1/2cos3t/2.sint/2

=(3/8).1/(cos³3t/2.sint/2

put t= pi/2

d²y/dx²=3/8. 1/(cos³(3pi/4).sin(pi/4))

=-3/2

Thanks & Regards

Rinkoo Gupta

AskIITians Faculty

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if x=2 cos t - cos 2t, y=2 sin t – sin 2t find d2y/dx2 at t=pi/2

if x=2 cos t - cos 2t, y=2 sin t – sin 2t find d2y/dx2 at t=pi/2

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