A-level · Trigonometry

The R-form (a sinθ + b cosθ)Exam-style questions

Writing a sinθ ± b cosθ as a single R sin(θ ± α) or R cos(θ ∓ α), finding R and α by matching coefficients, and using the form to read off maxima/minima, solve equations and answer modelling questions.

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23 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAllLevel 2Level 3Level 4Level 5
  1. 1
    Find the maximum value of 3sin⁡θ+4cos⁡θ3\sin\theta + 4\cos\theta and the smallest positive value of θ\theta (in degrees, to 1 d.p.) at which it occurs. State also the minimum value.
    Level 4 · 3 marksTry it
  2. 2
    (a) Express 3sin⁡θ+4cos⁡θ3\sin\theta + 4\cos\theta in the form Rsin⁡(θ+α)R\sin(\theta + \alpha), where R>0R > 0 and 0<α<π20 < \alpha < \dfrac{\pi}{2}. Give α\alpha correct to 3 significant figures.
    (b) Hence write down the maximum value of 3sin⁡θ+4cos⁡θ3\sin\theta + 4\cos\theta, and find the smallest positive value of θ\theta at which it occurs.
    Level 4 · 6 marksTry it
  3. 3
    (a) Express 5cos⁡x+12sin⁡x5\cos x + 12\sin x in the form Rcos⁡(x−α)R\cos(x - \alpha), where R>0R > 0 and 0<α<90∘0 < \alpha < 90^\circ. Give α\alpha to 2 decimal places. (3 marks)
    (b) Hence solve, for 0≤x<360∘0 \le x < 360^\circ, the equation

    5cos⁡x+12sin⁡x=6.55\cos x + 12\sin x = 6.5,

    giving your answers to 1 decimal place. (5 marks)
    (c) Write down the minimum value of 5cos⁡x+12sin⁡x5\cos x + 12\sin x, and the value of xx in 0≤x<360∘0 \le x < 360^\circ at which it occurs. (2 marks)
    (d) Explain why the equation 5cos⁡x+12sin⁡x=145\cos x + 12\sin x = 14 has no solutions. (2 marks)
    Level 4 · 12 marksTry it
  4. 4
    (a) Express 5cos⁡θ−12sin⁡θ5\cos\theta - 12\sin\theta in the form Rcos⁡(θ+α)R\cos(\theta + \alpha), where R>0R > 0 and 0∘<α<90∘0^{\circ} < \alpha < 90^{\circ}. Give α\alpha correct to 22 decimal places. (3 marks)
    (b) Hence solve the equation 5cos⁡θ−12sin⁡θ=45\cos\theta - 12\sin\theta = 4 for 0∘⩽θ⩽360∘0^{\circ} \leqslant \theta \leqslant 360^{\circ}. (3 marks)
    Level 4 · 6 marksTry it
  5. 5
    (a) By first expanding sin⁡(x+60∘)\sin(x + 60^{\circ}), express 4sin⁡(x+60∘)−5sin⁡x4\sin(x + 60^{\circ}) - 5\sin x in the form Rcos⁡(x+α)R\cos(x + \alpha), where R>0R > 0 and 0∘<α<90∘0^{\circ} < \alpha < 90^{\circ}. Give the value of RR correct to 4 significant figures and the value of α\alpha correct to 2 decimal places.

    (5 marks)
    (b) Hence solve the equation

    4sin⁡(x+60∘)−5sin⁡x=1,4\sin(x + 60^{\circ}) - 5\sin x = 1,

    for 0∘<x<360∘0^{\circ} < x < 360^{\circ}.

    (4 marks)
    Level 4 · 9 marksTry it

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