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
    Given that

    5sin⁡θ+3cos⁡θ≡Rsin⁡(θ+α)5\sin\theta + 3\cos\theta \equiv R\sin(\theta + \alpha)

    where R>0R > 0 and 0<α<π20 < \alpha < \dfrac{\pi}{2}, find RR exactly and find α\alpha, in radians, correct to 3 decimal places.

    (3 marks)
    Level 2 · 3 marksTry it
  2. 2
    Given that

    4sin⁡θ−7cos⁡θ≡Rsin⁡(θ−α)4\sin\theta - 7\cos\theta \equiv R\sin(\theta - \alpha)

    where R>0R > 0 and 0<α<90∘0 < \alpha < 90^\circ, find the value of RR and the value of α\alpha, giving α\alpha to 1 decimal place.

    (3 marks)
    Level 2 · 3 marksTry it
  3. 3
    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<90∘0 < \alpha < 90^\circ. Give α\alpha to 1 decimal place.
    Level 3 · 3 marksTry it
  4. 4
    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 < \alpha < 90^\circ. Give α\alpha to 1 decimal place.
    Level 3 · 3 marksTry it
  5. 5
    Express sin⁡θ−3cos⁡θ\sin\theta - \sqrt{3}\cos\theta in the form Rsin⁡(θ−α)R\sin(\theta - \alpha), where R>0R > 0 and α\alpha is acute, giving exact values.
    Level 3 · 3 marksTry it

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