Back to course
7.5PureStretch

Simplifying a cos x ± b sin x

Any expression asin⁡x±bcos⁡xa\sin x \pm b\cos x can be written as a single sine or cosine wave Rsin⁡(x±α)R\sin(x \pm \alpha). This “harmonic form” makes maxima, minima and solving equations easy.

30 min Trigonometry and Modelling
Edexcel A level Maths: 7.5 Simplifying a cos x + b sin x (R form)Curated by Kepler, not produced by us. Watch it before the notes below.
Open on YouTube

What you'll be able to do

  • Write a sin x ± b cos x as R sin(x ± α)
  • Find R and α
  • Use the form to find max/min values
  • Solve equations with the R-form
1

Finding R and α

Compare asin⁡x+bcos⁡x=Rsin⁡(x+α)=Rcos⁡αsin⁡x+Rsin⁡αcos⁡xa\sin x + b\cos x = R\sin(x+\alpha) = R\cos\alpha\sin x + R\sin\alpha\cos x. So Rcos⁡α=aR\cos\alpha = a and Rsin⁡α=bR\sin\alpha = b, giving R=a2+b2R = \sqrt{a^2 + b^2} and tan⁡α=ba\tan\alpha = \tfrac{b}{a}.

R=a2+b2,tan⁡α=baR = \sqrt{a^2 + b^2}, \qquad \tan\alpha = \tfrac{b}{a}
R is always positive; α is acute.
2

Maxima and minima

Since sin⁡\sin and cos⁡\cos range over [−1,1][-1,1], the expression Rsin⁡(x+α)R\sin(x+\alpha) has maximum RR and minimum −R-R. This gives instant max/min values.

Example 1Write 3sin⁡x+4cos⁡x3\sin x + 4\cos x as Rsin⁡(x+α)R\sin(x + \alpha).
1R=32+42=5R = \sqrt{3^2 + 4^2} = 5.
2tan⁡α=43⇒α≈53.13°\tan\alpha = \tfrac43 \Rightarrow \alpha \approx 53.13°.
Answer
5sin⁡(x+53.13°)5\sin(x + 53.13°); max 55, min −5-5.

Tip — Match the form to the target: a sin x + b cos x → R sin(x + α); a cos x + b sin x → R cos(x − α).

Formula recap

R=a2+b2R = \sqrt{a^2 + b^2}
Amplitude.
tan⁡α=ba\tan\alpha = \tfrac{b}{a}
Phase angle.
max⁡=R, min⁡=−R\max = R, \ \min = -R
From the single wave.

Common mistakes to avoid

Taking R negative.
R = √(a²+b²) is always positive.
Using the wrong target form.
Match sin/cos and sign carefully to set up R cos α and R sin α correctly.

Key takeaways

  • a sin x ± b cos x = R sin(x ± α).
  • R = √(a²+b²), tan α = b/a.
  • Max = R, min = −R.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. For asin⁡x+bcos⁡x=Rsin⁡(x+α)a\sin x + b\cos x = R\sin(x+\alpha), RR equals:

    • a2+b2\sqrt{a^2 + b^2}
    • a+ba + b
    • a2−b2\sqrt{a^2 - b^2}
    • abab
    Show answer and explanation

    Answer: a2+b2\sqrt{a^2 + b^2}

    Pythagoras on coefficients.

  2. tan⁡α\tan\alpha equals:

    • ba\tfrac{b}{a}
    • ab\tfrac{a}{b}
    • abab
    • a+ba + b
    Show answer and explanation

    Answer: ba\tfrac{b}{a}

    From Rsinα/Rcosα.

  3. 3sin⁡x+4cos⁡x3\sin x + 4\cos x has R=R =

    • 55
    • 77
    • 7\sqrt7
    • 1212
    Show answer and explanation

    Answer: 55

    √(9+16).

  4. The maximum value of 5sin⁡(x+α)5\sin(x + \alpha) is:

    • 55
    • 11
    • α\alpha
    • 1010
    Show answer and explanation

    Answer: 55

    R is the amplitude.

  5. R is always:

    • positive
    • negative
    • zero
    • an angle
    Show answer and explanation

    Answer: positive

    Square root taken positive.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

All 7 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.