7.5PureStretch

Simplifying a cos x ± b sin x

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

30 min Video by Zeeshan Zamurred Trigonometry and Modelling

Edexcel A level Maths: 7.5 Simplifying a cos x + b sin x (R form)Watch the full walkthrough before the notes below.

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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

Finding R and α

Compare $a sin x + b cos x = R sin (x + α) = R cos α sin x + R sin α cos x$ . So $R cos α = a$ and $R sin α = b$ , giving $R = a^{2} + b^{2}$ and $tan α = \frac{b}{a}$ .

R = a^{2} + b^{2}, tan α = \frac{b}{a}

R is always positive; α is acute.

Maxima and minima

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

R = 3^{2} + 4^{2} = 5

tan α = \frac{4}{3} \Rightarrow α \approx 53.13°

Answer

5 sin (x + 53.13°)

; max

5

, min

- 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 = a^{2} + b^{2}

Amplitude.

tan α = \frac{b}{a}

Phase angle.

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.

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