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Trigonometric Ratios and Identities · Paper 3 — Free Maths AI HL Practice Questions

1ChallengeLAQUnit circle and angle measurement10 marksPaper 3~15 min
A particle moves on a unit circle centred at the origin OO. Its position at time tt seconds is P ⁣(cos⁡θ(t), sin⁡θ(t))P\!\left(\cos\theta(t),\,\sin\theta(t)\right), where θ(t)\theta(t) is measured in radians from the positive xx-axis. The angular velocity is ω(t)=dθdt\omega(t)=\dfrac{d\theta}{dt}. The motion is defined by θ(t)=arcsin⁡(t)\theta(t)=\arcsin(t) for 0≤t≤10\leq t\leq 1.
(a)
Write down the velocity vector P˙(t)=ddt ⁣(cos⁡θ, sin⁡θ)\dot{P}(t)=\dfrac{d}{dt}\!\left(\cos\theta,\,\sin\theta\right) in terms of ω(t)\omega(t) and θ(t)\theta(t), and hence show that the speed of the particle is —P˙—=—ω(t)—— \dot{P} — = — \omega(t) — for all tt in the domain. [3 marks]
(b)
Show that cos⁡(θ(t))=1−t2\cos(\theta(t))=\sqrt{1-t^2} for 0≤t≤10\leq t\leq 1, and hence prove that ω(t)=11−t2\omega(t)=\dfrac{1}{\sqrt{1-t^2}}. [4 marks]
(c)
Using your result from (b), evaluate ∫01/2ω(t) dt\displaystyle\int_{0}^{1/2}\omega(t)\,dt and verify that this equals θ ⁣(12)−θ(0)\theta\!\left(\tfrac{1}{2}\right)-\theta(0). Hence interpret, in terms of arc length on the unit circle, what this integral represents. [3 marks]
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