Find the locations where the tangent of the following parametric curve is horizo

Question

Find the locations where the tangent of the following parametric curve is horizontal or vertical. It is helpful to first draw this curve using your calculator. x = sin 2t, y = sin t; 0 ? t ? 2?

Explanation / Answer

For x = sin ( 2t ) dx / dt = 2* cos ( 2t ) y = sin ( t ) dy/ dt = cos t therefore dy / dx = ( dy / dt ) / ( dy / dx ) = 2* cos( 2t ) / cos ( t) for tangent to be horizontal dy / dx = zero i.e. cos 2t = 0 => t = (2* n + 1 ) * pi / 4 where n = 0 , 1, 2 ,........ on putting t = pi / 4 , 3*pi / 4 , 5*pi/4........... therefore points ( x, y) = ( 1, sqrt ( 1/ 2) ), ( -1 , sqrt ( 1/ 2) ) for tangent to be vertical dy / dx = infinity i.e. cost = 0 => t = ( 2*m + 1 ) * pi / 2 where m = 0 , 1, 2, ............ on putting t = pi / 2 , 3*pi / 2, 5*pi/2........... therefore points ( x, y) = ( 0, 1 ), ( 0 , -1 )

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