The Cosine Rule

Not drawn accurately. Work out the length of \(BC\). [3 marks]

Mark scheme — 3 marks available

• \(x^2 = 3^2 + 8^2 - 2 \times 3 \times 8 \times \cos 60^\circ\) — M1
• \(9 + 64 - 24 = 49\) — M1
• \(7\) — A1

Model answer

A triangle has sides of length 3 cm, 5 cm and 7 cm. Work out the size of the largest angle. [3 marks]

Mark scheme — 3 marks available

• \(\cos A = \dfrac{3^2 + 5^2 - 7^2}{2 \times 3 \times 5}\) — M1
• \(-\dfrac{1}{2}\) — A1
• \(120\) — A1

Model answer

Not drawn accurately. Two ships leave a port \(P\) at the same time. Ship \(Q\) sails 30 km and ship \(R\) sails 50 km. The angle between their paths is \(120^\circ\). Work out the distance \(QR\) between the ships. [3 marks]

Mark scheme — 3 marks available

• \(x^2 = 30^2 + 50^2 - 2 \times 30 \times 50 \times \cos 120^\circ\) — M1
• \(900 + 2500 + 1500 = 4900\) — M1
• \(70\) — A1

Model answer

In triangle \(ABC\), \(AB = AC = 9\) cm and angle \(A = 120^\circ\). Work out the length of \(BC\). Give your answer as a surd in its simplest form. [3 marks]

Mark scheme — 3 marks available

• \(BC^2 = 9^2 + 9^2 - 2 \times 9 \times 9 \times \cos 120^\circ\) — M1
• \(243\) — M1
• \(9\sqrt{3}\) — A1

Model answer

A triangle has sides of length 7 cm, 8 cm and 13 cm. (a) Work out the size of the largest angle. [3 marks] (b) Work out the exact area of the triangle. [2 marks]

Mark scheme — 5 marks available

• (a) \(\cos A = \dfrac{7^2 + 8^2 - 13^2}{2 \times 7 \times 8}\) — M1
• (a) \(-\dfrac{1}{2}\) — A1
• (a) \(120\) — A1
• (b) \(\dfrac{1}{2} \times 7 \times 8 \times \sin 120^\circ\) — M1
• (b) \(14\sqrt{3}\) — A1

Model answer

A triangle has sides of length 5 cm, 5 cm and \(5\sqrt{3}\) cm. Work out the size of the largest angle. [3 marks]

Mark scheme — 3 marks available

• \(\cos A = \dfrac{5^2 + 5^2 - (5\sqrt{3})^2}{2 \times 5 \times 5}\) — M1
• \(-\dfrac{1}{2}\) — A1
• \(120\) — A1

Model answer

Which is the cosine rule for the side \(a\)?

1. A \(a^2 = b^2 + c^2 + 2bc\cos A\)
2. B \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B}\)
3. C \(a = b + c - 2bc\cos A\)
4. D \(a^2 = b^2 + c^2 - 2bc\cos A\) Correct

Why: The cosine rule takes \(2bc\cos A\) away from the sum of the squares.

When do you use the cosine rule to find a side?

1. A When you know two angles and a side
2. B When you know a side and its opposite angle
3. C When you know two sides and the angle between them Correct
4. D When the triangle has a right angle

Why: Two sides and the included angle is the cosine rule situation.

What is \(\cos 60^\circ\)?

1. A \(\dfrac{\sqrt{3}}{2}\)
2. B \(\dfrac{1}{2}\) Correct
3. C \(\dfrac{\sqrt{2}}{2}\)
4. D 1

Why: This is an exact value.

What does the cosine rule become when \(A = 90^\circ\)?

1. A Pythagoras' theorem Correct
2. B The sine rule
3. C The area formula
4. D \(a = b + c\)

Why: \(\cos 90^\circ = 0\), so \(a^2 = b^2 + c^2\).

In triangle \(ABC\), \(b = 3\), \(c = 8\) and \(A = 60^\circ\). What is \(a^2\)?

1. A 73
2. B 97
3. C 25
4. D 49 Correct

Why: \(a^2 = 9 + 64 - 2 \times 3 \times 8 \times \frac{1}{2} = 73 - 24 = 49\).

In triangle \(ABC\), \(b = 6\), \(c = 10\) and \(A = 120^\circ\). What is \(a\)?

1. A 8
2. B \(\sqrt{76}\)
3. C 14 Correct
4. D 16

Why: \(a^2 = 36 + 100 - 120 \times (-\frac{1}{2}) = 136 + 60 = 196\), so \(a = 14\).

A triangle has sides 5, 7 and 8. What is \(\cos\) of the angle opposite the side of length 7?

1. A \(-\dfrac{1}{2}\)
2. B \(\dfrac{1}{2}\) Correct
3. C \(\dfrac{1}{7}\)
4. D \(\dfrac{3}{5}\)

Why: \(\cos A = \dfrac{5^2 + 8^2 - 7^2}{2 \times 5 \times 8} = \dfrac{40}{80} = \dfrac{1}{2}\).

A triangle has sides 3, 5 and 7. What is the largest angle?

1. A \(120^\circ\) Correct
2. B \(60^\circ\)
3. C \(150^\circ\)
4. D \(90^\circ\)

Why: \(\cos A = \dfrac{9 + 25 - 49}{30} = -\dfrac{1}{2}\), so \(A = 120^\circ\).

A triangle has sides 7, 8 and 13. What is the angle opposite the longest side?

1. A \(60^\circ\)
2. B \(90^\circ\)
3. C \(150^\circ\)
4. D \(120^\circ\) Correct

Why: \(\cos A = \dfrac{49 + 64 - 169}{112} = \dfrac{-56}{112} = -\dfrac{1}{2}\), so \(A = 120^\circ\).