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Trigonometric Functions · Paper 2 — Free Maths AI HL Practice Questions

1MasterySAQ-STrigonometric identities and equations10 marksPaper 2~15 min
A water wheel of radius 88 metres rotates at a constant angular speed. The height, hh metres, of a point PP on the rim of the wheel above the water level is modelled by h(t)=asin⁡(bt)+dh(t) = a\sin(bt) + d where tt is the time in seconds. At t=0t = 0, point PP is at its lowest point, 22 metres above the water level. The wheel completes one full revolution every 6060 seconds.
(a)
Show that a=6a = 6, b=π30b = \dfrac{\pi}{30}, and d=10d = 10. [3 marks]
(b)
Calculate the first time t>0t > 0 when the height of PP is exactly 1111 metres above the water level. Give your answer correct to 3 significant figures. [3 marks]
(c)
A second point QQ on the rim is always diametrically opposite to PP. Its height is modelled by g(t)=6sin⁡ ⁣(π30t+π)+10g(t) = 6\sin\!\left(\dfrac{\pi}{30}t + \pi\right) + 10. Using the identity sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A + B) = \sin A \cos B + \cos A \sin B, show that g(t)=−6sin⁡ ⁣(π30t)+10g(t) = -6\sin\!\left(\dfrac{\pi}{30}t\right) + 10, and hence find all times in the interval 0≤t≤600 \leq t \leq 60 when PP and QQ are at the same height. [4 marks]
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