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Learn9709Paper 3 · Pure Mathematics 3TrigonometryThe R-form (harmonic form)
Trigonometry · Subtopic 4 of 4

The R-form: a sin θ + b cos θ

Any combination asin⁡θ+bcos⁡θa\sin\theta+b\cos\theta is a single sine wave in disguise: Rsin⁡(θ+α)R\sin(\theta+\alpha). How to find RR and α\alpha, and how this makes equations and maximum and minimum questions easy.

Cambridge 9709 Paper 2Cambridge 9709 Paper 3Edexcel YMA01 WMA13Edexcel 9MA0 Paper 1Edexcel 9MA0 Paper 2AQA 7357 Paper 1≈ 45 minRevision 6 · updated 2026-10-27
Section 1 of 14

Learning objectives

  • Express asin⁡θ+bcos⁡θa\sin\theta+b\cos\theta as Rsin⁡(θ±α)R\sin(\theta\pm\alpha) or Rcos⁡(θ±α)R\cos(\theta\pm\alpha).
  • Find R=a2+b2R=\sqrt{a^2+b^2} and α\alpha by comparing coefficients.
  • Solve equations of the form asin⁡θ+bcos⁡θ=ca\sin\theta+b\cos\theta=c.
  • Find greatest and least values of expressions involving asin⁡θ+bcos⁡θa\sin\theta+b\cos\theta.
Pause and think

Sketch or imagine y=3sin⁡x+4cos⁡xy=3\sin x+4\cos x. It is the sum of two waves with the same period. What shape do you expect? How high might it go?

Section 2 of 14

Core idea

Start hereTwo waves make one

Play two notes of the same pitch slightly out of step and you hear one note, just louder or softer. Mathematically, 3sin⁡θ+4cos⁡θ3\sin\theta+4\cos\theta is a single sine wave: 5sin⁡(θ+53.1∘)5\sin(\theta+53.1^\circ). Writing it this way instantly reveals its maximum (55) and makes equations easy.

The big idea

asin⁡θ+bcos⁡θ≡Rsin⁡(θ+α)a\sin\theta+b\cos\theta\equiv R\sin(\theta+\alpha) with R=a2+b2R=\sqrt{a^2+b^2} and tan⁡α=ba\tan\alpha=\frac ba. Other forms (Rcos⁡(θ±α)R\cos(\theta\pm\alpha), Rsin⁡(θ−α)R\sin(\theta-\alpha)) are found the same way: expand and match coefficients.

1. Deriving R and α

3 sin θ + 4 cos θ = 5 sin(θ + 53.1°)
50100150200250300350−6−4−22460θ (degrees)y3 sin θ4 cos θsum: amplitude 5max 5 at 36.9°
The sum has amplitude 32+42=5\sqrt{3^2+4^2}=5, and it is the sine wave shifted left by 53.1∘53.1^\circ.
a sin θ + 4 cos θ as a changes
50100150200250300350−6−4−22460θ (degrees)y
The amplitude is always a2+16\sqrt{a^2+16} and the wave shifts as aa changes, but it stays a pure sine wave.
asin⁡θ+bcos⁡θ=Rsin⁡(θ+α)a\sin\theta+b\cos\theta=R\sin(\theta+\alpha)
  1. Rsin⁡(θ+α)=Rcos⁡αsin⁡θ+Rsin⁡αcos⁡θR\sin(\theta+\alpha)=R\cos\alpha\sin\theta+R\sin\alpha\cos\theta
    Compound formula.
  2. Rcos⁡α=a,Rsin⁡α=bR\cos\alpha=a,\quad R\sin\alpha=b
    Match coefficients of sin⁡θ\sin\theta and cos⁡θ\cos\theta.
  3. R2cos⁡2α+R2sin⁡2α=a2+b2⇒R=a2+b2R^2\cos^2\alpha+R^2\sin^2\alpha=a^2+b^2\Rightarrow R=\sqrt{a^2+b^2}
    Square and add; cos⁡2+sin⁡2=1\cos^2+\sin^2=1.
  4. Rsin⁡αRcos⁡α=ba⇒tan⁡α=ba\frac{R\sin\alpha}{R\cos\alpha}=\frac ba\Rightarrow\tan\alpha=\frac ba
    Divide.
Worked example3cos⁡x−sin⁡x\sqrt3\cos x-\sin x as Rcos⁡(x+α)R\cos(x+\alpha)

Find RR and α\alpha (0<α<90∘0<\alpha<90^\circ).

Goal
RR, α\alpha.
Strategy
Expand the target form and match.
  1. Rcos⁡(x+α)=Rcos⁡xcos⁡α−Rsin⁡xsin⁡αR\cos(x+\alpha)=R\cos x\cos\alpha-R\sin x\sin\alpha
    Compound.
  2. Rcos⁡α=3,Rsin⁡α=1R\cos\alpha=\sqrt3,\quad R\sin\alpha=1
    Match.
  3. R=2,tan⁡α=13⇒α=30∘R=2,\quad\tan\alpha=\tfrac1{\sqrt3}\Rightarrow\alpha=30^\circ
    Square-add; divide.
Answer2cos⁡(x+30∘)2\cos(x+30^\circ).
✓ Checkx=0x=0: 3\sqrt3 and 2cos⁡30=32\cos30=\sqrt3 ✓.

2. Uses

Worked example3sin⁡θ+4cos⁡θ=23\sin\theta+4\cos\theta=2

Solve for 0≤θ≤360∘0\le\theta\le360^\circ.

Goal
All θ\theta.
Strategy
Write as 5sin⁡(θ+53.13∘)5\sin(\theta+53.13^\circ).
  1. sin⁡(θ+53.13∘)=0.4\sin(\theta+53.13^\circ)=0.4
    Divide by 55.
  2. 53.13∘≤θ+53.13∘≤413.13∘53.13^\circ\le\theta+53.13^\circ\le413.13^\circ
    Shift the range.
  3. θ+53.13∘=156.42∘, 383.58∘\theta+53.13^\circ=156.42^\circ,\ 383.58^\circ
    sin⁡−10.4=23.58∘\sin^{-1}0.4=23.58^\circ (too small); 180−23.58180-23.58; 23.58+36023.58+360.
Answerθ=103.3∘,330.4∘\theta=103.3^\circ,330.4^\circ.
✓ Checkθ=103.3\theta=103.3: 3(0.973)+4(−0.230)=2.92−0.92=2.03(0.973)+4(-0.230)=2.92-0.92=2.0 ✓.
Worked exampleMaximum and minimum of a related expression

Find the max and min of 107−(3sin⁡θ+4cos⁡θ)\dfrac{10}{7-(3\sin\theta+4\cos\theta)}.

Goal
Extreme values.
Strategy
The wave ranges over [−5,5][-5,5].
  1. 2≤7−5sin⁡(θ+α)≤122\le7-5\sin(\theta+\alpha)\le12
    Bounds of the denominator.
  2. 1012≤expression≤102\tfrac{10}{12}\le\text{expression}\le\tfrac{10}2
    Smaller denominator gives larger value.
AnswerMax 55, min 56\frac56.
✓ CheckDenominator always positive, so no division by zero ✓.
Common misconceptionSigns in α Always expand the requested form; don't just use tan⁡α=ba\tan\alpha=\frac ba blindly. For Rsin⁡(θ−α)R\sin(\theta-\alpha) the cos term gets a minus.
Pause and think

Maximum of 5cos⁡x−12sin⁡x5\cos x-12\sin x?

Summary

  • Expand, match, square-add, divide.
  • Max RR, min −R-R.
Key words
Harmonic form
A single wave form.
HistoryHarmonic motion. Christiaan Huygens built the first accurate pendulum clock in 1656, based on the regular swing that we now call simple harmonic motion. Any such motion can be written as asin⁡t+bcos⁡ta\sin t+b\cos t or equivalently Rsin⁡(t+α)R\sin(t+\alpha): RR is the size of the swing and α\alpha is where it starts.
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