Let f(x)=6sinxcosx+6cos2x−1f(x) = 6 \sin x \cos x + 6 \cos^2 x - 1f(x)=6sinxcosx+6cos2x−1.
Write f(x)f(x)f(x) in the form asin2x+bcos2x+ca \sin 2x + b \cos 2x + casin2x+bcos2x+c where a,b,a, b,a,b, and ccc are integers to be found.
Use the answer to part (a) to write f(x)f(x)f(x) in the form Rsin(2x+α)+cR \sin (2x + \alpha) + cRsin(2x+α)+c where R>0R > 0R>0 and 0<α<π20 < \alpha < \frac{\pi}{2}0<α<2π. Give the exact value of RRR and give the value of α\alphaα in radians to 3 significant figures.
Hence, or otherwise, (i) state the maximum value of f(x)f(x)f(x), (ii) find the second smallest positive value of xxx at which a maximum value of f(x)f(x)f(x) occurs. Give your answer to 3 significant figures.
Practise Edexcel A Level Maths Trigonometry and Modelling with exam-style questions for A Level Maths. 100 questions covering 7.1 Addition Formulae, 7.2 Double Angle Formulae, 7.3 Solving Trigonometric Equations, 7.4 Simplifying a cos x +- b sin x, 7.5 Proving Trigonometric Identities, and 7.6 Modelling with Trigonometric Functions, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.