An engineer is experimenting with a new machine that features two rotating metal arms. The first arm is 2 metres long, and can rotate clockwise about a pivot at the origin O, completing one full revolution in 2 seconds. At the end of the first arm, at the point P, is another pivot to which the second arm is attached. The second arm is 1 metre long and can rotate anticlockwise about P, completing one full revolution in 2 seconds. At the end of the second arm is the point Q The engineer sets the machine so that the first arm makes an angle of 2π/3 with the positive x-axis, and the second arm makes an angle of π/2 with the positive x-axis, as shown in the above diagram. At t = 0 she starts the machine, and both arms begin rotating. 

(a) Determine an expression in terms of cosine for xP(t), the x-coordinate (in meters) of the point P at any time t (in seconds). 

(b) Determine an expression for xQ(t), the x-coordinate (in meters) of the point Q at any time t (in seconds), writing answer as the sum of a cosine function and xP(t). 

(c) Simplify your expression for xQ(t) to the form xQ(t) = A sin(πt − θ) where A > 0 and θ ∈ (−π, π]

 

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