1.05nPureStretch

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 $R\sin(x \pm \alpha)$ . This “harmonic form” makes maxima, minima and solving equations easy.

30 min Trigonometry

The big picture

For $a\sin x+b\cos x=R\sin(x+\alpha)$ , compare coefficients: $R\cos\alpha=a$ , $R\sin\alpha=b$ . Use both signs to choose the quadrant of the phase angle. When solving, transform the whole interval for x into the corresponding interval for x plus alpha. Finding only the calculator’s inverse-sine value loses the second branch and possibly a periodic solution.

Edexcel A level Maths: 7.5 Simplifying a cos x + b sin x (R form)Curated by Kepler, not produced by us. Watch it, then work through 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

Finding R and α

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

R = \sqrt{a^2 + b^2}, \qquad \tan\alpha = \tfrac{b}{a}

R is always positive; α is acute.

Maxima and minima

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

Example 1Write

3\sin x + 4\cos x

R\sin(x + \alpha)

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

\tan\alpha = \tfrac43 \Rightarrow \alpha \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).

Find every solution after combining waves

For $a\sin x+b\cos x=R\sin(x+\alpha)$ , compare coefficients: $R\cos\alpha=a$ , $R\sin\alpha=b$ . Use both signs to choose the quadrant of the phase angle.

When solving, transform the whole interval for x into the corresponding interval for x plus alpha. Finding only the calculator’s inverse-sine value loses the second branch and possibly a periodic solution.

Example 1Solve

3\sin x+4\cos x=2.5

for

0\leq x<360^\circ

R=5

and

\alpha=\tan^{-1}(4/3)=53.1301^\circ

\sin(x+\alpha)=1/2

with

53.1301^\circ\leq x+\alpha<413.1301^\circ

3The shifted angles in that interval are

150^\circ

and

390^\circ

4Subtract

\alpha

to obtain

96.8699^\circ

and

336.8699^\circ

Answer

x=96.9^\circ,336.9^\circ

(1 d.p.).

Think like an examiner

Write the defining equations and show the substitution before evaluating.
Check the domain, units and interpretation of the final result.

Common misconceptions

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 results to remember

R = \sqrt{a^2 + b^2}

Amplitude.

\tan\alpha = \tfrac{b}{a}

Phase angle.

\max = R, \ \min = -R

From the single wave.

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.

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 10 lessons