AQA Level 2 Further Maths · Level 2 Certificate · Chapter 7: Geometry II · Lesson 2

The sine rule

About 90 minutesCalculator allowed66 marks

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Date

Do Now

Answer these before the lesson starts.

A1
GCSE Higher
In triangle $A B C$ , angle $B = 9 0^{\circ}$ , $A C = 17$ cm and angle $A = 2 9^{\circ}$ . Work out $B C$ to 3 significant figures.
[2 marks]
Answer $B C = \dots\dots$ cm
A2
Needed today
Solve $sin θ = 0.43$ for $0^{\circ} < θ < 18 0^{\circ}$ . Give your answers to 1 decimal place.
[2 marks]
Answer $θ = \dots\dots$ or $θ = \dots\dots$
A3
Last lesson
A triangle has sides of $11$ cm and $13$ cm with an angle of $4 7^{\circ}$ between them. Work out its area to 3 significant figures.
[2 marks]
Answer $\dots\dots$ cm²
A4
From chapter 6
Write down the exact values of $sin 6 0^{\circ}$ and $sin 13 5^{\circ}$ .
[2 marks]
Answer $sin 6 0^{\circ} = \dots\dots sin 13 5^{\circ} = \dots\dots$

Example, then your turn

Level 2

Your teacher works through each example with you. Then try the one beside it. Give lengths and angles to 3 significant figures unless the question says otherwise.

Example 1

Work out the length $x$ .

Not drawn accurately

Hint Pair each side with the angle opposite it:

\frac{a}{s i n A} = \frac{b}{s i n B}

. The angle opposite the

11

cm side is not given, so find it from the angle sum first.

B1
Work out the length $x$ .
[3 marks]
Not drawn accurately
Answer $x = \dots\dots$ cm

Example 2

Work out the size of angle $θ$ .

Not drawn accurately

Hint To find an angle, use the form

\frac{s i n A}{a} = \frac{s i n C}{c}

, with the angle on top.

B2
Work out the size of angle $θ$ .
[3 marks]
Not drawn accurately
Answer $θ = \dots \dots^{\circ}$

Example 3

In triangle $A B C$ , angle $B = 3 5^{\circ}$ , $A B = 10$ cm and $A C = 7$ cm. Work out the two possible sizes of angle $C$ .

Hint The diagram shows why there are two triangles: the

7

cm side can swing to meet the base at

C_{1}

or at

C_{2}

B3
In triangle $P QR$ , angle $Q = 2 8^{\circ}$ , $P Q = 11$ cm and $P R = 6.5$ cm. Work out the two possible sizes of angle $R$ .
[3 marks]
Answerangle $R = \dots \dots^{\circ}$ or $\dots \dots^{\circ}$

Example 4

A ship sails $18$ km from port $P$ on a bearing of $07 0^{\circ}$ to $Q$ . It then sails on a bearing of $15 0^{\circ}$ to $R$ , which is due east of $P$ . Work out the distance $P R$ .

Not drawn accurately

Hint Angle

QP R = 9 0^{\circ} - 7 0^{\circ}

. At

Q

, the bearing back to

P

25 0^{\circ}

, so angle

P QR = 25 0^{\circ} - 15 0^{\circ}

B4
Jo walks $6.4$ km from $A$ on a bearing of $05 0^{\circ}$ to $B$ . She then walks on a bearing of $30 0^{\circ}$ until she reaches $C$ , which is due north of $A$ . How far is she from $A$ ?
[4 marks]
Not drawn accurately
Answer $\dots\dots$ km

Practice

Level 2

C1
In the triangle, $k$ is a positive number. Use the sine rule to work out the size of the obtuse angle $x$ .
[3 marks]
Not drawn accurately
Answer $x = \dots \dots^{\circ}$
C2
$A B C D$ is a quadrilateral. Work out the length of $C D$ .
[4 marks]
Not drawn accurately
Answer $C D = \dots\dots$ cm

C3
$A B C D$ is a kite with $A B = A D$ and $C B = C D$ . The diagonal $A C$ is $15$ cm. Angle $B A D = 11 6^{\circ}$ and angle $B C D = 4 0^{\circ}$ . Work out the perimeter of the kite.
[3 marks]
Not drawn accurately
Answer $\dots\dots$ cm
C4
In triangle $A B C$ , angle $A = 4 0^{\circ}$ , $B C = 10$ cm and $A C = 7$ cm. Explain why angle $B$ has only one possible size, and work it out.
[3 marks]
Not drawn accurately
Answerangle $B = \dots \dots^{\circ}$

Exam-style questions

AQA exam style

D1
In triangle $P QR$ , angle $P = 4 2^{\circ}$ , $QR = 8$ cm and $P R = 11$ cm. Ravi says, "Angle $Q$ is $66. 9^{\circ}$ . There is only one triangle with these measurements."
(a)
Explain why Ravi is wrong.
[2 marks]
(b)
Work out the difference between the two possible lengths of $P Q$ .
[3 marks]
Answer $\dots\dots$ cm
Total for D1: 5 marks

D2
A boat is at $A$ . A lighthouse $L$ is on a bearing of $05 2^{\circ}$ from $A$ . The boat sails $12$ km due east to $B$ . From $B$ , the lighthouse is on a bearing of $33 4^{\circ}$ .
Not drawn accurately
(a)
Show that angle $A L B = 7 8^{\circ}$ .
[2 marks]
(b)
Work out the distance $B L$ .
[2 marks]
Answer $B L = \dots\dots$ km
(c)
The boat sails back towards $A$ . Work out the shortest distance between the boat and the lighthouse on this journey.
[2 marks]
Answer $\dots\dots$ km
Total for D2: 6 marks
D3
Two posts $A$ and $B$ stand $60$ m apart on one bank of a river. A tree $T$ is on the other bank. The banks are parallel. Angle $T A B = 6 3^{\circ}$ and angle $T B A = 4 8^{\circ}$ . Work out the width of the river. You must show your working.
[5 marks]
Not drawn accurately
Answer $\dots\dots$ m
D4
$D$ is a point on the side $B C$ of triangle $A B C$ . $A B = 9$ cm, angle $A B D = 5 8^{\circ}$ , angle $A D B = 7 4^{\circ}$ and angle $A C D = 4 0^{\circ}$ .
Not drawn accurately
(a)
Work out the length of $A D$ .
[2 marks]
Answer $A D = \dots\dots$ cm
(b)
Work out the length of $A C$ .
[2 marks]
Answer $A C = \dots\dots$ cm
(c)
Work out the area of triangle $A D C$ .
[2 marks]
Answer $\dots\dots$ cm²
Total for D4: 6 marks
D5
$Q$ is $5$ km due north of $P$ . Route 1 goes from $P$ on a bearing of $03 5^{\circ}$ to $X$ , then on a bearing of $32 0^{\circ}$ to $Q$ . Route 2 goes from $P$ on a bearing of $34 0^{\circ}$ to $Y$ , then on a bearing of $06 0^{\circ}$ to $Q$ . Which route is shorter, and by how much? Give your answer to 3 decimal places.
[6 marks]
Not drawn accurately
AnswerRoute $\dots\dots$ is shorter by $\dots\dots$ km

Extension

Stretch

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

E1
In triangle $A B C$ , angle $A = 3 5^{\circ}$ and $B C = 6$ cm. The length of $A C$ is $b$ cm. Find the set of values of $b$ for which there are two different triangles $A B C$ . Give the end values to 3 significant figures where needed.
[4 marks]
Hint 1 · What to try
Try $b = 5$ , $b = 8$ and $b = 12$ . For each, work out $sin B$ and decide how many triangles there are.
Hint 2 · The first line
$sin B = \frac{b sin 3 5 ^{\circ}}{6}$ . Two angles $B$ need $sin B < 1$ , and the obtuse one must fit with $A = 3 5^{\circ}$ .
Hint 3 · The full method
$sin B < 1$ gives $b < \frac{6}{sin 3 5 ^{\circ}} = 10.46$ . The obtuse $B = 18 0^{\circ} - B_{1}$ fits only if $18 0^{\circ} - B_{1} < 14 5^{\circ}$ , that is $B_{1} > 3 5^{\circ}$ , so $sin B > sin 3 5^{\circ}$ and $b > 6$ . So $6 < b < 10.5$ .
Answer $\dots\dots < b < \dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use the sine rule to find a side when I know two angles and a side.
I can use the sine rule to find an angle when I know two sides and an angle not between them.
I can tell when there are two possible triangles and find both answers.
I can use the sine rule in bearings and multi-step problems.

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