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

Double-angle formulae

113 marks

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Quick questions to start.

A1
Needed today
Use $cos (A + B) = cos A cos B - sin A sin B$ to write $cos (x + x)$ in terms of $cos x$ and $sin x$ .
[2 marks]
Answer
A2
Needed today
$cos θ = \frac{5}{13}$ and $θ$ is acute. Find the exact values of $sin θ$ and $tan θ$ .
[2 marks]
Answer
A3
Common slip
Show that $sin (2 \times 3 0^{\circ})$ is not equal to $2 sin 3 0^{\circ}$ .
[2 marks]
A4
From chapter 6
Solve $2 sin^{2} x - sin x - 1 = 0$ for $0^{\circ} \leq x < 36 0^{\circ}$ .
[3 marks]
Answer

A-Level

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

Example 1

$cos A = \frac{4}{5}$ and $A$ is acute. Find the exact values of $sin 2 A$ and $cos 2 A$ .

Hint Find sin A first, then use sin 2A = 2 sin A cos A.

B1
$sin A = - \frac{5}{13}$ and $π < A < \frac{3 π}{2}$ . Find the exact values of $sin 2 A$ , $cos 2 A$ and $tan 2 A$ .
[5 marks]
Answer

Example 2

Solve $sin 2 x = cos x$ for $0^{\circ} \leq x < 36 0^{\circ}$ .

Hint Write sin 2x as 2 sin x cos x and factorise. Do not divide by cos x.

B2
Solve $sin 2 x + sin x = 0$ for $0^{\circ} \leq x < 36 0^{\circ}$ .
[5 marks]
Answer

Example 3

Solve $cos 2 x = sin x$ for $0^{\circ} \leq x < 36 0^{\circ}$ .

Hint Choose the form of cos 2x that leaves only sin x.

Example 4

Show that $2 sin^{2} x = 1 - cos 2 x$ . Hence find the exact value of $sin^{2} 1 5^{\circ}$ .

Hint Start from cos 2x = 1 − 2 sin²x.

B4
Show that $cos^{4} x - sin^{4} x = cos 2 x$ . Hence find the exact value of $cos^{4} \frac{π}{12} - sin^{4} \frac{π}{12}$ .
[4 marks]

Example 5

$tan A = \frac{1}{2}$ . Find the exact value of $tan 2 A$ .

Hint Use tan 2A = 2 tan A divided by (1 − tan²A).

B5
$A$ is acute and $tan 2 A = \frac{3}{4}$ . Find the exact value of $tan A$ , and the exact value of $tan 4 A$ .
[5 marks]
Answer

A-Level

C1
Leo solves $sin 2 x = 2 sin x$ for $0^{\circ} \leq x < 36 0^{\circ}$ like this: $2 sin x cos x = 2 sin x, cos x = \frac{2}{2}, x = 4 5^{\circ}, 31 5^{\circ} .$ Explain the mistake, and give all the solutions.
[4 marks]
Answer

C2
Use $cos 2 A = 2 cos^{2} A - 1$ with $A = 22. 5^{\circ}$ to find the exact value of $cos 22. 5^{\circ}$ . Find the exact value of $sin 22. 5^{\circ}$ in the same way.
[4 marks]
Answer
C3
Write $cos 3 x$ as $cos (2 x + x)$ and prove that $cos 3 x = 4 cos^{3} x - 3 cos x$ .
[5 marks]
C4
Show that $\frac{1 - cos 2 x}{sin 2 x} = tan x$ . Hence find the exact value of $tan 1 5^{\circ}$ .
[5 marks]
C5
Solve $cos 2 x = 5 sin x - 2$ for $0^{\circ} \leq x < 36 0^{\circ}$ .
[4 marks]
Answer

Edexcel exam style

D2
Ava writes: " $cos 2 x = 1 - 2 sin x$ , so $1 - 2 sin x + sin x = 0$ , which gives $sin x = 1$ and $x = 9 0^{\circ}$ only."
Ava is asked to solve $cos 2 x + sin x = 0$ for $0^{\circ} \leq x < 36 0^{\circ}$ .
(a)
Explain the mistake in Ava's working.
[2 marks]
(b)
Solve the equation correctly.
[5 marks]
Answer
Total for D2: 7 marks
D3
Solve $2 sin 2 θ = 3 tan θ$ for $0^{\circ} \leq θ < 36 0^{\circ}$ .
[7 marks]
Answer
D4
In triangle $A B C$ , $A B = A C = 6$ cm and angle $B A C = 2 θ$ , where $θ$ is acute.
(a)
Show that the area of the triangle is $18 sin 2 θ$ cm².
[2 marks]
(b)
Show that $B C = 12 sin θ$ cm.
[3 marks]
(c)
Given that $cos θ = \frac{4}{5}$ , find the exact area of the triangle and the exact length of $B C$ .
[4 marks]
Answer
Total for D4: 9 marks
D5
(a)
Show that $\frac{sin 2 x}{1 + cos 2 x} = tan x$ , for $cos 2 x \neq = - 1$ .
[3 marks]
(b)
Hence find the exact value of $tan 67. 5^{\circ}$ .
[3 marks]
Answer
(c)
Hence solve $\frac{sin 2 x}{1 + cos 2 x} = 3$ for $0^{\circ} < x < 18 0^{\circ}$ , $x \neq = 9 0^{\circ}$ .
[3 marks]
Answer
Total for D5: 9 marks
D6
(a)
Show that $8 sin^{4} x = 3 - 4 cos 2 x + cos 4 x$ .
[5 marks]
(b)
Hence find the exact value of $sin^{4} \frac{π}{8}$ .
[2 marks]
Answer
(c)
State the greatest value of $3 - 4 cos 2 x + cos 4 x$ and the value of $x$ in $0 \leq x < π$ where it occurs.
[2 marks]
Answer
Total for D6: 9 marks

Stretch

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

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find sin 2A, cos 2A and tan 2A from one ratio of A, choosing signs from the quadrant.
I can choose the form of cos 2A that leaves one trigonometric function, and solve the equation.
I can use cos 2A to find exact values of half angles such as cos 22.5°.
I can rewrite powers such as sin⁴x in terms of cos 2x and cos 4x, and prove identities.

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