The temperature, θ\thetaθ °C, inside a warehouse is modelled by the equation
θ=8+2sin(15t−160)°+3cos(15t−160)°\theta = 8 + 2\sin(15t - 160)° + 3\cos(15t - 160)°θ=8+2sin(15t−160)°+3cos(15t−160)°
where t t\,t is the number of hours after midnight.
Express 2sinx+3cosx2\sin x + 3\cos x2sinx+3cosx in the form Rsin(x+α)R\sin(x + \alpha)Rsin(x+α), where R>0 R > 0\,R>0 and 0°<α<90°0° < \alpha < 90°0°<α<90°. Give R R\,R to three significant figures and α \alpha\,α to two decimal places.
Deduce the maximum temperature inside the warehouse during the day.
Find the time of day at which this maximum temperature occurs, giving your answer to the nearest minute.
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.