AQA Level 2 Further Maths · Level 2 Certificate · Chapter 7: Geometry II · Lesson 3

The cosine rule, and using both rules

About 90 minutesCalculator allowed65 marks

Hints online, answers free with an account:
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Date

Do Now

Answer these before the lesson starts.

A1
Needed today
Solve $x^{2} - 5 x - 14 = 0$ .
[2 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$
A2
Needed today
Solve $cos θ = - 0.35$ for $0^{\circ} < θ < 18 0^{\circ}$ . Give your answer to 1 decimal place.
[2 marks]
Answer $θ = \dots \dots^{\circ}$
A3
Last lesson
In triangle $A B C$ , angle $A = 5 2^{\circ}$ , angle $B = 7 1^{\circ}$ and $B C = 8.6$ cm. Work out $A C$ to 3 significant figures.
[2 marks]
Answer $A C = \dots\dots$ cm
A4
From chapter 7
A triangle has sides of $6$ cm and $10$ cm with an angle of $15 0^{\circ}$ between them. Work out its exact area.
[2 marks]
Answer $\dots\dots$ cm²

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 You know two sides and the angle between them, so use

a^{2} = b^{2} + c^{2} - 2 b c cos A

. Remember to take the square root at the end. If the angle is obtuse,

cos A

is negative and the last term is added.

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 You know three sides, so use

cos A = \frac{b ^{2} + c ^{2} - a ^{2}}{2 b c}

, where

a

is the side opposite the angle you want. The largest angle is opposite the longest side.

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

Example 3

Work out the length $x$ . Do not use a calculator.

Not drawn accurately

Hint The unknown

x

is next to the

6 0^{\circ}

angle, so put

x

into the cosine rule for the side opposite:

1 3^{2} = x^{2} + 7^{2} - 2 \times x \times 7 \times cos 6 0^{\circ}

. This gives a quadratic.

B3
Work out the length $x$ . Do not use a calculator.
[4 marks]
Not drawn accurately
Answer $x = \dots\dots$ cm

Example 4

A ship sails $12$ km from harbour $H$ on a bearing of $04 0^{\circ}$ to $A$ . It then sails $9$ km on a bearing of $11 5^{\circ}$ to $B$ . Work out the distance $H B$ and the bearing of $B$ from $H$ .

Not drawn accurately

Hint At

A

, the bearing back to

H

22 0^{\circ}

, so angle

H A B = 22 0^{\circ} - 11 5^{\circ}

. Use the cosine rule for

H B

, then the sine rule for angle

A H B

B4
Mia walks $5.2$ km from $A$ on a bearing of $30 0^{\circ}$ to $B$ . She then walks $7.5$ km on a bearing of $06 0^{\circ}$ to $C$ . Work out the distance $A C$ and the bearing of $C$ from $A$ .
[5 marks]
Not drawn accurately
Answer $A C = \dots\dots$ km, bearing $\dots \dots^{\circ}$

Practice

Level 2

C1
In triangle $A B C$ , $A B = 5$ cm, $B C = 8$ cm and $cos B = \frac{1}{5}$ . Work out the exact length of $A C$ .
[3 marks]
Not drawn accurately
Answer $A C = \dots\dots$ cm
C2
A field is a triangle with sides of $42$ m, $55$ m and $68$ m. Work out its area to the nearest square metre.
[4 marks]
Not drawn accurately
Answer $\dots\dots$ m²

C3
$A B C D$ is a parallelogram. The diagonals $A C = 22$ cm and $B D = 14$ cm meet at $O$ . The angle between the diagonals is $6 4^{\circ}$ . Work out the lengths of the sides of the parallelogram.
[4 marks]
Not drawn accurately
Answer $\dots\dots$ cm and $\dots\dots$ cm
C4
A circle with centre $A$ and radius $6$ cm and a circle with centre $B$ and radius $9$ cm meet at $P$ . $A B = 11$ cm. Work out the size of angle $P A B$ .
[3 marks]
Not drawn accurately
Answerangle $P A B = \dots \dots^{\circ}$

Exam-style questions

AQA exam style

D1
In triangle $A B C$ , $A B = 8.5$ cm, $A C = 6$ cm and angle $B A C = 5 2^{\circ}$ . Sam works out $B C$ like this: " $B C^{2} = (8. 5^{2} + 6^{2} - 2 \times 8.5 \times 6) \times cos 5 2^{\circ} = 3.85$ , so $B C = 1.96$ cm."
Not drawn accurately
(a)
Explain the mistake Sam has made.
[1 mark]
(b)
Work out the length of $B C$ .
[2 marks]
Answer $B C = \dots\dots$ cm
(c)
Work out the size of angle $A B C$ .
[2 marks]
Answer $\dots \dots^{\circ}$
Total for D1: 5 marks

D2
$P QR S$ is a quadrilateral. Angle $P QR = 9 0^{\circ}$ , $P Q = 9$ cm, $QR = 12$ cm, $R S = 10$ cm and $S P = 8$ cm.
Not drawn accurately
(a)
Show that $P R = 15$ cm.
[1 mark]
(b)
Work out the size of angle $P S R$ .
[3 marks]
Answer $\dots \dots^{\circ}$
(c)
Work out the area of $P QR S$ .
[3 marks]
Answer $\dots\dots$ cm²
Total for D2: 7 marks
D3
Two boats leave port $P$ at 12 noon. Boat $A$ sails on a bearing of $03 5^{\circ}$ at $14$ km/h. Boat $B$ sails on a bearing of $11 0^{\circ}$ at $20$ km/h.
Not drawn accurately
(a)
Work out the distance between the boats at 1.30 pm.
[3 marks]
Answer $\dots\dots$ km
(b)
Work out the bearing of $B$ from $A$ at 1.30 pm. You must show your working.
[3 marks]
Answer $\dots \dots^{\circ}$
Total for D3: 6 marks
D4
In triangle $A B C$ , $A B = (4 + 3)$ cm, $A C = (4 - 3)$ cm and $B C = 5$ cm. Do not use a calculator in this question.
Not drawn accurately
(a)
Show that angle $B A C = 6 0^{\circ}$ .
[4 marks]
(b)
Work out the area of the triangle. Give your answer in the form $\frac{a 3}{b}$ where $a$ and $b$ are integers.
[2 marks]
Answer $\dots\dots$ cm²
Total for D4: 6 marks

Extension

Stretch

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

E1
In triangle $A B C$ , $A B = 7$ cm, $A C = 9$ cm and $B C = 10$ cm. $M$ is the midpoint of $B C$ . Work out the exact length of $A M$ .
[4 marks]
Hint 1 · What to try
$A M$ is a side of triangle $A B M$ . You know $A B$ and $B M$ ; you need the angle at $B$ .
Hint 2 · The first line
In triangle $A B C$ , $cos B = \frac{7 ^{2} + 1 0 ^{2} - 9 ^{2}}{2 \times 7 \times 10} = \frac{68}{140}$ .
Hint 3 · The full method
In triangle $A B M$ , $A M^{2} = 7^{2} + 5^{2} - 2 \times 7 \times 5 \times \frac{68}{140} = 74 - 34 = 40$ , so $A M = 40 = 210$ cm.
Answer $A M = \dots\dots$ cm

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use the cosine rule to find a side when I know two sides and the angle between them.
I can use the cosine rule to find an angle when I know three sides.
I can decide whether to use the sine rule or the cosine rule.
I can use both rules in bearings, area and multi-step problems.

Answers and mark scheme

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