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Maths · More trigonometry

Calculating areas and the sine rule

Use the labelling of triangles, area \(= \tfrac{1}{2}ab\sin C\), and the sine rule to find missing sides and angles in non-right-angled triangles.

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Teacher resources

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Student handouts

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Warm-up

Answer each one, then check.

  1. 1

    What is the area of a triangle?

    Show answerHide answer

    \(\tfrac{1}{2} \times \text{base} \times \text{height}\)

  2. 2

    What do the angles in a triangle add up to?

    Show answerHide answer

    \(180^\circ\)

  3. 3

    What is \(\sin^{-1}(0.5)\)?

    Show answerHide answer

    \(30^\circ\)

  4. 4

    Which side is opposite the right angle?

    Show answerHide answer

    The hypotenuse

  5. 5

    What does SOH CAH TOA help with?

    Show answerHide answer

    Right-angled triangles only

Learning Objectives

  1. 1Label a triangle with lower-case sides opposite capital angles.
  2. 2Find the area of any triangle using \(\tfrac{1}{2}ab\sin C\).
  3. 3Use the sine rule to find a missing side.
  4. 4Use the sine rule to find a missing angle.

SINE RULE

In any triangle, the sides and the sines of their opposite angles are in proportion.

\(\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}\). Flip it to find an angle: \(\dfrac{\sin A}{a} = \dfrac{\sin B}{b}\).

Which Formula?

  • Area

    \(\tfrac{1}{2}ab\sin C\): two sides and the angle between them.

  • Sine rule (side)

    Two angles and a side, need another side.

  • Sine rule (angle)

    Two sides and a non-included angle, need an angle.

  • Neither

    Three sides, or two sides and the included angle: use the cosine rule (next lesson).

Area of a Triangle

Find the area of a triangle with sides \(a = 8\) cm and \(b = 11\) cm and angle \(C = 50^\circ\) between them. Give your answer to 3 significant figures.

Show the solutionHide the solution
  1. 1 Write the formula Area \(= \tfrac{1}{2}ab\sin C\)
  2. 2 Substitute \(\tfrac{1}{2} \times 8 \times 11 \times \sin 50^\circ\)
  3. 3 Work out \(44 \times 0.7660 = 33.7\)

AnswerArea \(= 33.7\) cm\(^2\)

Sine Rule: Finding a Side

In triangle ABC, \(A = 40^\circ\), \(B = 65^\circ\) and \(a = 8\) cm. Find \(b\).

Show the solutionHide the solution
  1. 1 Set up \(\dfrac{b}{\sin 65^\circ} = \dfrac{8}{\sin 40^\circ}\)
  2. 2 Rearrange \(b = \dfrac{8 \times \sin 65^\circ}{\sin 40^\circ}\)
  3. 3 Work out \(b = 11.28\)

Answer\(b = 11.3\) cm (3 s.f.)

Sine Rule: Finding an Angle

In triangle ABC, \(a = 7\) cm, \(b = 9\) cm and angle \(A = 35^\circ\). Find angle \(B\).

Show the solutionHide the solution
  1. 1 Set up \(\dfrac{\sin B}{9} = \dfrac{\sin 35^\circ}{7}\)
  2. 2 Rearrange \(\sin B = \dfrac{9 \times \sin 35^\circ}{7} = 0.7375\)
  3. 3 Inverse sine \(B = \sin^{-1}(0.7375) = 47.5^\circ\)

Answer\(B = 47.5^\circ\) (1 d.p.)

Setting Out

Put the unknown on top.

  • Missing side

    Write the rule with sides on top: \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B}\).

  • Missing angle

    Write the rule with sines on top: \(\dfrac{\sin A}{a} = \dfrac{\sin B}{b}\).

  • Diagram

    Mark which side is opposite which angle before you start.

  • Check

    The biggest angle is opposite the longest side.

Choose the Method

For each, say whether you would use the area formula, the sine rule, or neither yet, and why. (a) Two angles and one side, find another side. (b) Two sides and the angle between them, find the area. (c) Three sides, find an angle. (d) Two sides and an angle not between them, find another angle.

1. List what is given.

2. Choose the formula.

A good answer shows: (a) Sine rule. (b) Area \(= \tfrac{1}{2}ab\sin C\). (c) Neither yet: cosine rule. (d) Sine rule.

Can I...?

  1. 1Label sides and angles.
  2. 2Write the area formula.
  3. 3Find the area of a triangle.
  4. 4Write the sine rule.
  5. 5Find a missing side.
  6. 6Find a missing angle.
  7. 7Put the unknown on top.
  8. 8Check the answer is sensible.

Summary & Exam Focus

  • Area \(= \tfrac{1}{2}ab\sin C\).
  • Sine rule: \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B}\).
  • Use the sine rule with two angles and a side, or two sides and a non-included angle.
  • Check side lengths against angles.

Exam focus

In triangle ABC, \(AB = 7\) cm, angle \(B = 40^\circ\) and angle \(C = 75^\circ\). Work out the length of \(AC\). Give your answer correct to 3 significant figures. (3 marks) (3 marks)

Write the sine rule with the unknown side on top. Keep your calculator in degrees.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Sine rule
\(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\).
Opposite
Across the triangle from an angle, not touching it.
Included angle
The angle between two given sides.
Scalene
A triangle with no equal sides.
Vertex
A corner of a shape.
Non-right-angled
A triangle with no \(90^\circ\) angle.

Questions and answers

12 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Work out 3 marks Easier

In triangle ABC, \(AB = 7\) cm, angle \(ABC = 40^\circ\) and angle \(ACB = 75^\circ\). Work out the length of \(AC\). Give your answer correct to 3 significant figures.

Triangle ABC with AB equal to 7 cm, angle B equal to 40 degrees and angle C equal to 75 degrees, with side AC to be found.

Mark scheme — 3 marks available

  • \(\dfrac{AC}{\sin 40^\circ} = \dfrac{7}{\sin 75^\circ}\) — M1
  • \(\dfrac{7 \times \sin 40^\circ}{\sin 75^\circ}\) — M1
  • 4.66 — A1

Model answer

\(AC = \dfrac{7 \sin 40^\circ}{\sin 75^\circ} = 4.66\) cm

2. Exam question Work out 3 marks Easier

In triangle ABC, angle \(A = 52^\circ\), angle \(B = 71^\circ\) and \(BC = 9.4\) cm. Work out the length of \(AC\). Give your answer correct to 3 significant figures.

Mark scheme — 3 marks available

  • \(\dfrac{AC}{\sin 71^\circ} = \dfrac{9.4}{\sin 52^\circ}\) — M1
  • \(\dfrac{9.4 \times \sin 71^\circ}{\sin 52^\circ}\) — M1
  • 11.3 — A1

Model answer

\(AC = \dfrac{9.4 \sin 71^\circ}{\sin 52^\circ} = 11.3\) cm

3. Exam question Work out 3 marks Easier

In triangle ABC, \(BC = 7\) cm, \(AC = 9\) cm and angle \(BAC = 35^\circ\). Work out the size of angle \(ABC\). Give your answer correct to 1 decimal place. (Angle \(ABC\) is acute.)

Mark scheme — 3 marks available

  • \(\dfrac{\sin B}{9} = \dfrac{\sin 35^\circ}{7}\) — M1
  • \(\sin B = 0.7375...\) — M1
  • 47.5 — A1

Model answer

\(\sin B = \dfrac{9 \sin 35^\circ}{7} = 0.7375\), so \(B = 47.5^\circ\)

4. Exam question Work out 2 marks Easier

A triangle has two sides of length 9.5 cm and 6.2 cm with an angle of \(112^\circ\) between them. Work out the area of the triangle. Give your answer correct to 3 significant figures.

Mark scheme — 2 marks available

  • \(\tfrac{1}{2} \times 9.5 \times 6.2 \times \sin 112^\circ\) — M1
  • 27.3 — A1

Model answer

\(\tfrac{1}{2} \times 9.5 \times 6.2 \times \sin 112^\circ = 27.3\) cm\(^2\)

5. Exam question Work out 4 marks Easier

The area of a triangle is 25 cm\(^2\). Two of its sides are 5 cm and 10 cm and the angle between them is acute. Work out the size of the angle between these two sides. Give your answer correct to 1 decimal place.

Mark scheme — 4 marks available

  • \(\tfrac{1}{2} \times 5 \times 10 \times \sin C = 25\) — M1
  • \(25 \sin C = 25\) — M1
  • \(\sin C = 1\) — A1
  • 90.0 — A1

Model answer

\(\tfrac{1}{2} \times 5 \times 10 \times \sin C = 25\), so \(\sin C = 1\), giving \(C = 90^\circ\).

6. Exam question Explain 3 marks Easier

In a triangle, angle \(A = 30^\circ\), \(a = 6\) cm and \(b = 10\) cm. Show that \(\sin B = \dfrac{5}{6}\).

Mark scheme — 3 marks available

  • Sine rule set up — M1
  • \(\sin 30^\circ = 0.5\) — M1
  • Completes the proof — A1

Model answer

\(\dfrac{\sin B}{10} = \dfrac{\sin 30^\circ}{6}\), so \(\sin B = \dfrac{10 \times 0.5}{6} = \dfrac{5}{6}\).

7. Multiple choice 1 mark Easier

In triangle ABC, side \(a\) is opposite...

  1. A Angle A Correct
  2. B Angle B
  3. C Angle C
  4. D The right angle

Why: Lower-case a is the side opposite capital A.

8. Multiple choice 1 mark Core

The area of a triangle with sides \(a\), \(b\) and included angle \(C\) is...

  1. A \(ab\sin C\)
  2. B \(\tfrac{1}{2}ab\cos C\)
  3. C \(\tfrac{1}{2}ab\sin C\) Correct
  4. D \(\tfrac{1}{2}a\sin C\)

Why: \(\tfrac{1}{2}ab\sin C\).

9. Multiple choice 1 mark Core

Which lets you find a side using the sine rule?

  1. A Three sides
  2. B Two angles and a side Correct
  3. C Two sides and the angle between them
  4. D Three angles

Why: Two angles and a side.

10. Multiple choice 1 mark Core

To find an angle with the sine rule, write it as...

  1. A \(\dfrac{a}{\sin A}\) with a on top
  2. B \(a \sin A\)
  3. C \(\dfrac{a}{b}\) only
  4. D \(\dfrac{\sin A}{a} = \dfrac{\sin B}{b}\) Correct

Why: Sines on top: \(\dfrac{\sin A}{a} = \dfrac{\sin B}{b}\).

11. Multiple choice 1 mark Core

A triangle has sides 4 and 6 with an angle of \(30^\circ\) between them. Its area is...

  1. A 6 Correct
  2. B 12
  3. C 3
  4. D 24

Why: \(\tfrac{1}{2} \times 4 \times 6 \times 0.5 = 6\).

12. Multiple choice 1 mark Stretch

In a triangle, the longest side is opposite...

  1. A The smallest angle
  2. B The right angle
  3. C The largest angle Correct
  4. D Any angle

Why: The largest angle.