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The R-formula (a cos + b sin)AQA A-Level Maths: Flashcards

What these 13 flashcards ask

  • Write a\cos\theta+b\sin\theta as R\cos(\theta-\alpha): what are R and \tan\alpha?
  • What equations do you get by expanding R\cos(\theta-\alpha)?
  • Express 3\cos\theta+4\sin\theta as R\cos(\theta-\alpha).
  • Express 5\sin x+12\cos x as R\sin(x+\alpha).
  • Express \sqrt3\sin\theta-\cos\theta as R\sin(\theta-\alpha).
  • Maximum value of a\cos\theta+b\sin\theta?
  • Minimum value of a\cos\theta+b\sin\theta?
  • Greatest value of 10+3\cos\theta+4\sin\theta?
  • When does R\cos(\theta-\alpha) first reach its maximum?
  • When does a\cos\theta+b\sin\theta=c have no solutions?
  • How do you solve a\cos\theta+b\sin\theta=c?
  • For 0\le\theta<360^\circ, what is the range of \theta-\alpha?
  • What is R in a model D=k+a\cos\omega t+b\sin\omega t?

Exam questions on The R-formula (a cos + b sin)

  1. Let f(θ)=3cos⁡θ+4sin⁡θf(\theta)=3\cos\theta+4\sin\theta, with θ\theta in degrees. It is written in the form rcos⁡(θ−α)r\cos(\theta-\alpha), where r>0r>0 and 0∘<α<90∘0^\circ<\alpha<90^\circ.
    Write down the maximum value of f(θ)f(\theta) and the smallest positive value of θ\theta at which it occurs.2 marks
  2. Let g(θ)=3sin⁡θ−cos⁡θg(\theta)=\sqrt3\sin\theta-\cos\theta, with θ\theta in degrees. It is written in the form Rsin⁡(θ−α)R\sin(\theta-\alpha), where R>0R>0 and 0∘<α<90∘0^\circ<\alpha<90^\circ.
    Solve g(θ)=1g(\theta)=1 for 0∘≤θ≤360∘0^\circ\le\theta\le360^\circ.2 marks
  3. Let h(x)=5sin⁡x+12cos⁡xh(x)=5\sin x+12\cos x, where xx is in degrees.
    Express h(x)h(x) in the form Rsin⁡(x+α)R\sin(x+\alpha), where R>0R>0 and 0∘<α<90∘0^\circ<\alpha<90^\circ.3 marks
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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).