The angular displacement of a pendulum, Θ\ThetaΘ, is modelled by the function Θ(t)=cost\Theta(t) = \cos tΘ(t)=cost, where ttt is the time in seconds measured from the equilibrium position. A specific displacement ppp is recorded at time t=ϕt = \phit=ϕ, where 0<ϕ<π20 < \phi < \frac{\pi}{2}0<ϕ<2π, such that cosϕ=p\cos \phi = pcosϕ=p.
State, in terms of ppp, the value of: (i) 10cos(2π−ϕ)10 \cos (2\pi - \phi)10cos(2π−ϕ) (ii) cos(ϕ−π)\cos (\phi - \pi)cos(ϕ−π) (iii) 0.5+cos(−ϕ)0.5 + \cos (-\phi)0.5+cos(−ϕ)
Sketch the graph of y=cos2ty = \cos 2ty=cos2t for the interval 0≤t≤π0 \le t \le \pi0≤t≤π, labelling the coordinates of the points where the curve meets the axes and any stationary points.
Determine, in terms of ϕ\phiϕ, the ttt-coordinates of any points in the interval 0<t<π0 < t < \pi0<t<π for which cos2t=p\cos 2t = pcos2t=p.
