The vertical displacement of a specialized laboratory sensor, VVV, is modelled by the function V(θ)=8cosθ+15sinθV(\theta) = 8\cos \theta + 15\sin \thetaV(θ)=8cosθ+15sinθ, where θ\thetaθ is the phase angle in radians.
Express V(θ)V(\theta)V(θ) in the form Rcos(θ−α)R\cos(\theta - \alpha)Rcos(θ−α), where R>0R > 0R>0 and 0<α<π20 < \alpha < \frac{\pi}{2}0<α<2π. Give the exact value of RRR and the value of α\alphaα, in radians, to 3 decimal places.
A secondary performance metric for the sensor, H(t)H(t)H(t), is defined by H(t)=12−3V(4t)H(t) = 12 - 3V(4t)H(t)=12−3V(4t), for t≥0t \ge 0t≥0, where ttt is the time in seconds.
Using the answer to part (a), (i) determine the exact maximum value of H(t)H(t)H(t). (ii) find the smallest positive value of ttt for which this maximum value occurs, giving your answer to 2 decimal places.
317 exam-style questions on AQA A Level Maths 1.8 E: Trigonometry, covering 1.8.1 Trigonometric definitions, rules and radians, 1.8.2 Small angle approximations (A-level only), 1.8.3 Trigonometric functions and exact values, 1.8.4 Reciprocal and inverse trigonometric functions (A-level only), 1.8.5 Trigonometric identities, 1.8.6 Compound and double angle formulae (A-level only), 1.8.7 Solving trigonometric equations, 1.8.8 Proofs with trigonometric identities, and 1.8.9 Trigonometry in context. Each one has a worked solution and a mark scheme showing where the marks go.