limjunyoung
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The first one is related with Cycloid
every 12seconds it completes a rotation, so every second it rotate 2pi/12=pi/6 radians
Radius=8cm
at t=0 h=16cm
after t seconds, the angle suspended by the raidus and the vertical is t*pi/6 radians
So the height is h(t)=16-[8-8cos(t*pi/6)]=8+8cos(t*pi/6)
The second one is related with SHM.
Convert the values of maximum and minimum so that the origin on a displacement time graph represents the depth of (18+4)/2= 11 meters
so the maximum is 18-11=7 minimum is 4-11=-7
x=Acos(a+wt)
at t=0 x=7 (since it's at maximum)
so x=7cos(wt) where w=2pi/f
at t=6.5 x=-7
so -7=7cos(6.5w) 6.5w=pi w=pi/6.5
so at any given time , x=7cos(t*pi/6.5)
the height h=x+11. so height h=11+7cos(t*pi/6.5)
Trig identity....
sin2x=2sinx*cosx
cos2x=1-2(sinx)^2 1-cos2x=2(sinx)^2
so sinx2x/(1-cos2x)=2sinx*cosx/2(sinx)^2=cosx/sinx=cotx
cotx+tanx=cosx/sinx +sinx/cosx=(cosx^2+sinx^2)/sinxcosx
=1/sinxcosx=2/2sinxcosx=2/sin2x=2cosec2x
so cotx=2cosec2x-tanx
hence establishes the result..
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