A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 7: Trigonometry and modelling · Lesson 4

Writing a cos x ± b sin x as R cos(x ± α)

119 marks

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Date

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Quick questions to start.

A1
Needed today
Expand $R cos (x - α)$ .
[2 marks]
Answer
A2
From chapter 6
The angle $α$ is acute and $tan α = \frac{3}{4}$ . Find $α$ to 1 decimal place and the exact value of $cos α$ .
[2 marks]
Answer
A3
Common slip
Find the greatest value of $5 - 2 cos x$ , and the value of $x$ in $0^{\circ} \leq x < 36 0^{\circ}$ where it happens.
[2 marks]
Answer
A4
Today's idea
$R cos α = 8$ and $R sin α = 15$ , where $R > 0$ and $α$ is acute. Find $R$ , and $α$ to 1 decimal place.
[3 marks]
Answer

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

Write $3 cos x + sin x$ in the form $R cos (x - α)$ , where $R > 0$ and $0^{\circ} < α < 9 0^{\circ}$ .

Hint Expand R cos(x − α) and match the coefficients of cos x and of sin x.

B1
Write $5 cos x + 12 sin x$ in the form $R cos (x - α)$ , where $R > 0$ and $0^{\circ} < α < 9 0^{\circ}$ . Give $α$ to 2 decimal places.
[5 marks]
Answer

Example 2

Write $2 sin x - 3 cos x$ in the form $R sin (x - α)$ , where $R > 0$ and $0^{\circ} < α < 9 0^{\circ}$ .

Hint R sin(x − α) = R sin x cos α − R cos x sin α. The minus sign is already in the form, so R cos α and R sin α are both positive.

B2
Write $4 sin x - cos x$ in the form $R sin (x - α)$ , where $R > 0$ and $0^{\circ} < α < 9 0^{\circ}$ . Give $R$ exactly and $α$ to 2 decimal places.
[5 marks]
Answer

Example 3

Find the greatest and least values of $3 cos x + 4 sin x$ , and the values of $x$ in $0^{\circ} \leq x < 36 0^{\circ}$ where they happen.

Hint Write it as 5 cos(x − 53.13°). A cosine is greatest when its angle is 0° and least when its angle is 180°.

B3
$f (x) = \frac{20}{15 - 5 c o s x - 12 s i n x}$ . Use your answer to B1 to find the greatest value of $f (x)$ , and the smallest positive value of $x$ where it happens.
[5 marks]
Answer

Example 4

Solve $3 cos x + sin x = 1$ for $0^{\circ} \leq x < 36 0^{\circ}$ .

Hint Use 2 cos(x − 30°) from Example 1. x − 30° runs from −30° to 330°.

B4
Solve $3 sin x - 4 cos x = 2$ for $0^{\circ} \leq x < 36 0^{\circ}$ . Give your answers to 1 decimal place.
[5 marks]
Answer

Example 5

Solve $sin 2 θ + cos 2 θ = 1$ for $0 \leq θ < π$ .

Hint Write it as √2 sin(2θ + π/4). Then 2θ + π/4 runs from π/4 to 9π/4.

B5
Solve $2 cos 2 θ - 3 sin 2 θ = 1$ for $0 \leq θ < π$ . Give your answers to 3 significant figures.
[5 marks]
Answer

Practice

A-Level

C1
Leo writes $6 cos x + 8 sin x = 10 cos (x + 53. 1^{\circ})$ . Explain his mistake and write the expression correctly.
[4 marks]

C2
$g (x) = 7 + 23 cos x - 2 sin x$ . Find the range of $g$ , and the smallest positive value of $x$ for which $g (x)$ is greatest.
[5 marks]
Answer
C3
State the number of solutions of $3 cos x + 4 sin x = k$ for $0^{\circ} \leq x < 36 0^{\circ}$ when (i) $k = 6$ (ii) $k = 5$ (iii) $k = 2$ . Give a reason.
[4 marks]
Answer
C4
Solve $8 cos x + 15 sin x = 10$ for $- 18 0^{\circ} < x < 18 0^{\circ}$ . Give your answers to 1 decimal place.
[5 marks]
Answer
C5
Write $cos θ + 3 sin θ$ in the form $R cos (θ - α)$ , where $R > 0$ and $0 < α < \frac{π}{2}$ , with $α$ exact. Hence find the least value of the expression and the value of $θ$ in $0 \leq θ < 2 π$ where it happens.
[4 marks]
Answer

Exam-style questions

Edexcel exam style

D1
(a)
Express $7 cos x - 24 sin x$ in the form $R cos (x + α)$ , where $R > 0$ and $0^{\circ} < α < 9 0^{\circ}$ . Give $α$ to 2 decimal places.
[3 marks]
Answer
(b)
Hence find the greatest value of $7 cos x - 24 sin x + 3$ , and the smallest positive value of $x$ at which it happens.
[3 marks]
Answer
(c)
Solve $7 cos x - 24 sin x = 15$ for $0^{\circ} \leq x < 36 0^{\circ}$ , giving your answers to 1 decimal place.
[4 marks]
Answer
Total for D1: 10 marks

D2
Priya writes: "The greatest value of $sin x$ is $1$ and the greatest value of $cos x$ is $1$ , so the greatest value of $3 sin x + 4 cos x$ is $3 + 4 = 7$ ."
Priya is asked for the greatest value of $3 sin x + 4 cos x$ .
(a)
Explain why Priya is wrong.
[2 marks]
(b)
Express $3 sin x + 4 cos x$ in the form $R sin (x + α)$ , where $R > 0$ and $0^{\circ} < α < 9 0^{\circ}$ . Give $α$ to 2 decimal places.
[3 marks]
Answer
(c)
$h (x) = (3 sin x + 4 cos x)^{2} + 6 (3 sin x + 4 cos x)$ . Find the greatest and least values of $h (x)$ .
[4 marks]
Answer
Total for D2: 9 marks
D3
Solve $sin x + 3 cos x = 2$ for $0 \leq x < 2 π$ . Give exact answers.
[7 marks]
Answer
D4
(a)
Show that $4 sin^{2} x + 6 sin x cos x$ can be written as $2 - 2 cos 2 x + 3 sin 2 x$ .
[3 marks]
(b)
Express $3 sin 2 x - 2 cos 2 x$ in the form $R sin (2 x - α)$ , where $R > 0$ and $0^{\circ} < α < 9 0^{\circ}$ . Give $R$ exactly and $α$ to 2 decimal places.
[3 marks]
Answer
(c)
Hence state the greatest value of $4 sin^{2} x + 6 sin x cos x$ , and solve $4 sin^{2} x + 6 sin x cos x = 3$ for $0^{\circ} \leq x < 18 0^{\circ}$ , giving your answers to 1 decimal place.
[5 marks]
Answer
Total for D4: 11 marks
D5
A rectangular panel $A B C D$ has $A B = 6$ m and $A D = 4$ m. Corner $A$ rests on horizontal ground and $A B$ makes an angle $θ$ with the ground, where $0^{\circ} < θ < 9 0^{\circ}$ . The panel is in a vertical plane. $h$ m is the height of $C$ above the ground. The figure is not drawn to scale.
(a)
Show that $h = 6 sin θ + 4 cos θ$ .
[2 marks]
(b)
Express $h$ in the form $R sin (θ + α)$ , where $R > 0$ and $0^{\circ} < α < 9 0^{\circ}$ . Give $R$ exactly and $α$ to 2 decimal places.
[3 marks]
Answer
(c)
Find the greatest possible height of $C$ , and the value of $θ$ that gives it.
[2 marks]
Answer
(d)
Find the values of $θ$ for which $h = 7$ , giving your answers to 1 decimal place.
[3 marks]
Answer
Total for D5: 10 marks
D6
$f (x) = \frac{12}{3 + cos x - 3 sin x}$ , $0^{\circ} \leq x < 36 0^{\circ}$ .
(a)
Express $cos x - 3 sin x$ in the form $R cos (x + α)$ , where $R > 0$ and $0^{\circ} < α < 9 0^{\circ}$ .
[2 marks]
Answer
(b)
Find the range of $f$ .
[3 marks]
Answer
(c)
Solve $f (x) = 4$ .
[3 marks]
Answer
Total for D6: 8 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Find the greatest and least values of $5 sin^{2} x + 6 sin x cos x - 3 cos^{2} x$ , and the values of $x$ in $0 \leq x < π$ where they happen. Give the values of $x$ to 3 significant figures.
[8 marks]
Hint 1 · What to try
Make every term a function of $2 x$ : use the double-angle formulae for $sin^{2} x$ , $cos^{2} x$ and $sin x cos x$ .
Hint 2 · The first line
You should get $1 - 4 cos 2 x + 3 sin 2 x$ . Write $3 sin 2 x - 4 cos 2 x$ as one sine.
Hint 3 · The full method
$5 sin (2 x - 0.927)$ is greatest when its angle is $\frac{π}{2}$ and least when it is $\frac{3 π}{2}$ , since $2 x - 0.927$ runs from $- 0.927$ to $2 π - 0.927$ .

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can write a cos x + b sin x as R cos(x − α) or R sin(x + α), with the right sign of α.
I can find R from R² = a² + b² and α from tan α, choosing the quadrant from R cos α and R sin α.
I can give the greatest and least values of an expression and the x where they happen.
I can solve a cos x + b sin x = c by moving the interval for x ± α first.

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