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

01 / LEARN

Understand the idea.

Match coefficients

To combine equal-frequency sine and cosine terms, expand $R sin (x + ϕ)$ and match coefficients. The sine coefficient is $R cos ϕ$ and the cosine coefficient is $R sin ϕ$ , so squaring and adding gives $R = \sqrt{a^{2} + b^{2}}$ .

Choose the phase using both coefficient signs: they determine the signs of $cos ϕ$ and $sin ϕ$ . Inverse tangent alone can choose the wrong quadrant. If $a = b = 0$ , the varying part is zero and no phase is needed.

a sin x + b cos x = R sin (x + ϕ)

Read the resulting range and equations

When $R > 0$ and the input spans all real values, the shifted sine attains both $- 1$ and $1$ , so its range is $[- R, R]$ . Adding a constant translates both endpoints without changing the amplitude.

The amplitude is not the sum of the individual coefficient magnitudes: sine and cosine peak at different inputs. After combining, solve equations using both sine branches and the phase shift. This method requires the original terms to share a frequency.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Example 1

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

Worked solution

1
Expand the desired form and match coefficients.
$\begin{aligned} R sin (x + ϕ) & = R cos ϕ sin x + R sin ϕ cos x \\ R cos ϕ & = 3, R sin ϕ = 4 \end{aligned}$
Matching the independent sine and cosine terms ensures equality for every real input.
2
Calculate the positive amplitude.
$R^{2} = 3^{2} + 4^{2} = 25, R = 5$
The identity ${cos}^{2} ϕ + {sin}^{2} ϕ = 1$ combines the coefficient equations.
3
Choose the acute phase.
$cos ϕ = \frac{3}{5}, sin ϕ = \frac{4}{5}, ϕ = {tan}^{- 1} (\frac{4}{3})$
Both ratios are positive, so the required phase lies in quadrant I.
4
Write the combined sinusoid.
$3 sin x + 4 cos x = 5 sin (x + {tan}^{- 1} (\frac{4}{3}))$
Expansion recovers the original coefficient pair exactly.

Answer

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

EXAMPLE 02

Example 2

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

Worked solution

1
Calculate the resultant amplitude.
$R = \sqrt{5^{2} + {(- 12)}^{2}} = \sqrt{169} = 13$
The two same-frequency terms combine into a sinusoid of this amplitude.
2
Bound the varying part.
$- 13 \leq 5 sin x - 12 cos x \leq 13$
A phase-shifted sine attains both extremes as real x varies.
3
Translate by the constant.
$- 7 \leq 6 + 5 sin x - 12 cos x \leq 19$
Adding six to every term shifts both attained endpoints, giving range $[- 7, 19]$ .

Answer

$[- 7, 19]$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

17.

The same-frequency terms have resultant amplitude $\sqrt{8^{2} + 15^{2}} = \sqrt{289} = 17$ .

Keep practising

6 worksheet questions · 12 marks

Open worksheet 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 $- \frac{π}{2} < ϕ < 0$ .
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 $\sqrt{2} (\frac{\sqrt{2}}{2}) = 1$ .
$\sqrt{2} sin (x + \frac{π}{4})$ .
Write $4 sin x - 3 cos x$ as $R sin (x + ϕ)$ with $- \frac{π}{2} < ϕ < 0$ .
2 marks
$R = 5$ , cosine of phase $\frac{4}{5}$ and sine of phase $- \frac{3}{5}$ .
$5 sin (x - {tan}^{- 1} (\frac{3}{4}))$ .
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 $\sqrt{2} sin (x + \frac{π}{4}) = 0$ , subtract $\frac{π}{4}$ and restrict the interval.
$\frac{3 π}{4}, \frac{7 π}{4}$ .
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.

Checking saved progress…

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.

Official curriculum source ↗

In this lesson

Understand the idea.

Match coefficients

Read the resulting range and equations

See the method in action.

Example 1

Expand the desired form and match coefficients.

Calculate the positive amplitude.

Choose the acute phase.

Write the combined sinusoid.

Example 2

Calculate the resultant amplitude.

Bound the varying part.

Translate by the constant.

Try it yourself.

Check your understanding

Keep practising