A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 7: Trigonometry and modelling · Lesson 1

Addition formulae

114 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/a-level-addition-formulae/

Date

Do Now

Quick questions to start.

A1
Needed today
Write down the exact values of $sin 3 0^{\circ}$ , $cos 4 5^{\circ}$ and $tan 6 0^{\circ}$ .
[3 marks]
Answer
A2
Needed today
$sin θ = \frac{3}{5}$ and $θ$ is obtuse. Find the exact values of $cos θ$ and $tan θ$ .
[3 marks]
Answer
A3
Common slip
Show that $sin (3 0^{\circ} + 6 0^{\circ})$ is not equal to $sin 3 0^{\circ} + sin 6 0^{\circ}$ .
[2 marks]
A4
From chapter 6
Solve $tan x = - 1$ for $0^{\circ} \leq x < 36 0^{\circ}$ .
[2 marks]
Answer

Example, then your turn

A-Level

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

Example 1

Use $cos (A + B) = cos A cos B - sin A sin B$ to find the exact value of $cos 7 5^{\circ}$ .

Hint Write 75° as a sum of two angles whose sines and cosines you know.

B1
Find the exact values of $sin 1 5^{\circ}$ and $cos 10 5^{\circ}$ .
[4 marks]
Answer

Example 2

Find the exact value of $tan 1 5^{\circ}$ , in the form $a + b 3$ .

Hint Use tan(A − B) = (tan A − tan B)/(1 + tan A tan B). The sign in the bottom is the opposite of the sign on top.

B2
Find the exact value of $tan 10 5^{\circ}$ , in the form $a + b 3$ .
[4 marks]
Answer

Example 3

$sin A = \frac{4}{5}$ with $A$ acute, and $cos B = - \frac{5}{13}$ with $B$ obtuse. Find the exact values of $sin (A + B)$ and $cos (A - B)$ .

Hint Find cos A and sin B first. In the second quadrant sine is positive.

B3
$sin A = \frac{8}{17}$ with $A$ acute, and $tan B = - \frac{3}{4}$ with $9 0^{\circ} < B < 18 0^{\circ}$ . Find the exact values of $cos (A + B)$ and $tan (A - B)$ .
[6 marks]
Answer

Example 4

Solve $sin (x + 3 0^{\circ}) = 2 cos x$ for $0^{\circ} \leq x < 36 0^{\circ}$ .

Hint Expand the left side, then collect the sin x terms on one side and the cos x terms on the other.

B4
Solve $cos (x - 6 0^{\circ}) = 3 sin x$ for $0^{\circ} \leq x < 36 0^{\circ}$ .
[5 marks]
Answer

Example 5

Show that $sin (A + B) + sin (A - B) = 2 sin A cos B$ . Hence find the exact value of $sin 7 5^{\circ} cos 1 5^{\circ}$ .

Hint Expand both, and the cos A sin B terms cancel. Then choose A and B so that A + B and A − B are angles you know.

B5
Prove that $cos (A + B) cos (A - B)$ is equal to $cos^{2} A - sin^{2} B$ . Hence find the exact value of $cos^{2} 7 5^{\circ} - sin^{2} 4 5^{\circ}$ .
[6 marks]
Answer

Practice

A-Level

C1
Mia works out $tan 7 5^{\circ}$ like this: $tan 7 5^{\circ} = tan (4 5^{\circ} + 3 0^{\circ}) = \frac{1 + \frac{1}{3}}{1 + \frac{1}{3}} = 1.$ Explain the mistake, and find the exact value of $tan 7 5^{\circ}$ in the form $a + b 3$ .
[4 marks]
Answer

C2
Write each as a single trigonometric function: (i) $cos 5 x cos 2 x + sin 5 x sin 2 x$ (ii) $\frac{tan 3 θ + tan θ}{1 - tan 3 θ tan θ}$ . Hence find the exact value of $cos \frac{5 π}{12} cos \frac{π}{12} + sin \frac{5 π}{12} sin \frac{π}{12}$ .
[4 marks]
Answer
C3
$tan (x + y) = 3$ and $tan x = \frac{1}{2}$ . Find the exact value of $tan y$ .
[4 marks]
Answer
C4
Show that $sin (x + \frac{π}{6}) - sin (x - \frac{π}{6}) = cos x$ . Hence solve $2 sin (x + \frac{π}{6}) - 2 sin (x - \frac{π}{6}) = 3$ for $0 \leq x < 2 π$ .
[5 marks]
C5
$A$ and $B$ are acute angles with $tan A = \frac{1}{2}$ and $tan B = \frac{1}{3}$ . Show that $A + B = \frac{π}{4}$ .
[4 marks]

Exam-style questions

Edexcel exam style

D1
(a)
Show that $sin (x + 3 0^{\circ}) + cos (x + 6 0^{\circ}) = cos x$ .
[3 marks]
(b)
Hence solve $sin (2 θ + 3 0^{\circ}) + cos (2 θ + 6 0^{\circ}) = - \frac{1}{2}$ for $0^{\circ} \leq θ < 36 0^{\circ}$ .
[4 marks]
Answer
(c)
Hence find the exact value of $sin 10 5^{\circ} + cos 13 5^{\circ}$ .
[2 marks]
Answer
Total for D1: 9 marks

D2
Sam writes: " $tan (A + B) = \frac{3 + 2}{1 - 6} = - 1$ , so $A + B = - 4 5^{\circ}$ . That is impossible for two acute angles, so no such angles exist."
$A$ and $B$ are acute angles with $tan A = 3$ and $tan B = 2$ .
(a)
Explain Sam's mistake, and state the size of $A + B$ .
[2 marks]
(b)
Find the exact value of $cos (A - B)$ , with a rational denominator.
[4 marks]
Answer
(c)
Find the exact value of $tan (A - B)$ .
[2 marks]
Answer
Total for D2: 8 marks
D3
Solve $2 sin (x + \frac{π}{4}) = cos (x - \frac{π}{4})$ for $0 \leq x < 2 π$ .
[6 marks]
Answer
D4
A wall stands on level ground. Point $P$ is on the ground, $d$ metres from the foot $F$ of the wall. Marks $L$ and $H$ are on the wall, $2$ m and $9$ m above $F$ . The angle $L P H$ is $θ$ .
(a)
When $d = 6$ , show that $tan θ = \frac{7}{9}$ .
[3 marks]
(b)
Show that $tan θ = \frac{7 d}{d ^{2} + 18}$ for any $d > 0$ .
[3 marks]
(c)
Hence find the two distances $d$ for which $tan θ = \frac{7}{11}$ .
[3 marks]
Answer
Total for D4: 9 marks
D5
(a)
Given that $sin (x + α) = 2 sin (x - α)$ , where $cos x cos α \neq = 0$ , show that $tan x = 3 tan α$ .
[4 marks]
(b)
Hence solve $sin (x + \frac{π}{6}) = 2 sin (x - \frac{π}{6})$ for $0 \leq x < 2 π$ .
[3 marks]
Answer
(c)
Solve $sin (x + \frac{π}{2}) = 2 sin (x - \frac{π}{2})$ for $0 \leq x < 2 π$ , explaining why part (a) cannot be used.
[2 marks]
Answer
Total for D5: 9 marks
D6
(a)
Prove that $cos (A + B) + cos (A - B) = 2 cos A cos B$ .
[2 marks]
(b)
Hence solve $cos 3 x + cos x = 0$ for $0 \leq x \leq π$ .
[4 marks]
Answer
(c)
Hence find the exact value of $cos 7 5^{\circ} + cos 1 5^{\circ}$ .
[3 marks]
Answer
Total for D6: 9 marks

Extension

Stretch

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

E1
The three angles of a triangle are $A$ , $B$ and $C$ , and $tan A$ , $tan B$ and $tan C$ are all positive whole numbers. Find them.
[8 marks]
Hint 1 · What to try
Since $A + B + C = 18 0^{\circ}$ , $tan C = - tan (A + B)$ . Expand and clear the fraction.
Hint 2 · The first line
You should reach $tan A + tan B + tan C = tan A tan B tan C$ . Call the three whole numbers $x \leq y \leq z$ .
Hint 3 · The full method
$x y z = x + y + z \leq 3 z$ , so $x y \leq 3$ . Check each small case.

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use sin(A ± B), cos(A ± B) and tan(A ± B) to find exact values such as cos 75° and tan 15°.
I can find sin(A + B) or cos(A − B) from given ratios, choosing the signs from the quadrant.
I can write an expression such as cos 5x cos 2x + sin 5x sin 2x as a single trigonometric function.
I can expand both sides of an equation to solve it, and prove results that use the addition formulae.

Answers and mark scheme

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.

Get the answers