Expressing a sine/cosine combination as one sinusoid | Year 11

01 / LEARN

The idea, step by step

Match coefficients

Expand $R sin (x + ϕ)$ to get $R cos ϕ sin x + R sin ϕ cos x$ . For $a sin x + b cos x$ , match $R cos ϕ = a$ and $R sin ϕ = b$ , so $R = \sqrt{a^{2} + b^{2}} \geq 0$ .

The phase must match both coefficient signs. Inverse tangent alone may select the wrong quadrant; use the signs of cosine and sine to choose a consistent angle. If both original coefficients are zero, the varying part is identically zero: its amplitude is zero and no phase needs to be chosen. Divide by the resultant amplitude only when it is positive.

a sin x + b cos x = R sin (x + ϕ)

Read the resulting range and equations

The combined function has range $[- R, R]$ . A vertical constant shifts the range but leaves the amplitude unchanged. Adding the individual amplitudes is incorrect because sine and cosine do not reach their maxima together.

To solve an equation, divide by $R$ , find both sine branches and subtract the phase. Check the required interval. This method assumes the two original terms have the same frequency; terms in $x$ and $2 x$ cannot generally be combined this way.

A common trap

Claiming $3 sin x + 4 cos x$ has amplitude 7.

Instead: Its amplitude is $\sqrt{3^{2} + 4^{2}} = 5$ , because the separate peaks occur at different angles.

02 / WORKED EXAMPLES

See the thinking

Example 1

Express $3 sin x + 4 cos x$ as $R sin (x + ϕ)$ with $R > 0$ and acute $ϕ$ .

$R = \sqrt{9 + 16} = 5$ .
Coefficient matching gives $cos ϕ = 3 / 5$ , $sin ϕ = 4 / 5$ .
Thus $ϕ = {tan}^{- 1} (4 / 3) \approx 0.9273$ .

5 sin (x + {tan}^{- 1} (4 / 3))

Example 2

Find the range of $6 + 5 sin x - 12 cos x$ .

The resultant amplitude is $\sqrt{25 + 144} = 13$ .
The two trigonometric terms together range from $- 13$ to 13.
Translate both endpoints up 6.

[- 7, 19]

Check your understanding

What is the amplitude of $8 sin x + 15 cos x$ ?

Show answer

17.

$\sqrt{64 + 225} = 17$ .

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find the amplitude of $sin x + cos x$ .
2 marks
Write $sin x + cos x$ as a single sine.
2 marks
Write $4 sin x - 3 cos x$ as $R sin (x + ϕ)$ with $- π / 2 < ϕ < 0$ .
2 marks
Find the range of $2 + \sqrt{3} sin x + cos x$ .
2 marks
Solve $sin x + cos x = 0$ on $[0, 2 π)$ .
2 marks
Why does the method not generally combine $a sin x + b cos 2 x$ into one sine?
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find the amplitude of $sin x + cos x$ .
2 marks
Take the positive square root of $1^{2} + 1^{2}$ .
$\sqrt{2}$ .
Write $sin x + cos x$ as a single sine.
2 marks
Both matched coefficients are $\sqrt{2} (\sqrt{2} / 2) = 1$ .
$\sqrt{2} sin (x + π / 4)$ .
Write $4 sin x - 3 cos x$ as $R sin (x + ϕ)$ with $- π / 2 < ϕ < 0$ .
2 marks
$R = 5$ , cosine of phase $4 / 5$ and sine of phase $- 3 / 5$ .
$5 sin (x - {tan}^{- 1} (3 / 4))$ .
Find the range of $2 + \sqrt{3} sin x + cos x$ .
2 marks
Amplitude is $\sqrt{3 + 1} = 2$ around midline 2.
$[0, 4]$ .
Solve $sin x + cos x = 0$ on $[0, 2 π)$ .
2 marks
Use $\sqrt{2} sin (x + π / 4) = 0$ , subtract $π / 4$ and restrict the interval.
$3 π / 4, 7 π / 4$ .
Why does the method not generally combine $a sin x + b cos 2 x$ into one sine?
2 marks
The expansion of a shifted sine produces sine and cosine terms with the same argument frequency.
The frequencies differ.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics, Further Trigonometry: periodic functions and models, trigonometric identities and reciprocal functions; the Library sequence follows the approved school-topic structure.

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