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2D Trigonometry

2D Trigonometry

The sine and cosine rules

Non Right - Sine Rule Extended 8 min read

On this page
  1. In this note
  2. Formula bank
  3. Labelling any triangle
  4. The sine rule
  5. The cosine rule
  6. Ways they test it
  7. Mistakes to avoid
  8. Summary
  9. Practice
  10. Can I do this?
  11. Quiz

In this note

Syllabus 6.3: the sine and cosine rules
  • 6.3Use the sine rule to find a side or an angle in any triangle.
  • 6.3Use the cosine rule to find a side or an angle in any triangle.
  • 6.3Handle obtuse angles and the ambiguous case.

Before you start: Right-angled trigonometry. Next: Area of a triangle.

Formula bank

Sine rule (side)On the formula sheet\(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\)a side over the sine of its OPPOSITE angle.Use it when a side and its opposite angle are known.
Sine rule (angle)Learn it\(\dfrac{\sin A}{a}=\dfrac{\sin B}{b}\)flip it when an angle is wanted.Use it when an angle wanted, opposite pair known.
Cosine rule (side)On the formula sheet\(a^{2}=b^{2}+c^{2}-2bc\cos A\)A is the angle BETWEEN sides b and c.Use it when two sides and the angle between them.
Cosine rule (angle)Learn it\(\cos A=\dfrac{b^{2}+c^{2}-a^{2}}{2bc}\)a is the side opposite the angle you want.Use it when all three sides known.
Obtuse sineLearn it\(\sin(180^\circ-x)=\sin x\)so a sine gives two angles: x and 180° − x.Use it when the ambiguous case.

Labelling any triangle

Triangle ABC: side a is opposite angle A, b opposite B, c opposite CABCabc
Small letter = the side opposite the capital-letter angle.

The right-angle rules (SOH CAH TOA) need a right angle. For ANY triangle use the sine rule or the cosine rule. Label the angles A, B, C and the side opposite each with the same small letter.

General methodHow to choose the rule

  1. Mark what you know and what you want.
  2. Do you know a side AND the angle opposite it (a full 'pair')? Use the SINE rule.
  3. No full pair? Two sides and the angle between them, or all three sides: use the COSINE rule.
  4. Need more? Find one more piece, then repeat (angles in a triangle add to 180°).

The sine rule

General methodHow to use the sine rule

  1. Write the pair you know and the part you want as two fractions.
  2. Side wanted: \(\dfrac{b}{\sin B}=\dfrac{a}{\sin A}\), multiply across.
  3. Angle wanted: \(\dfrac{\sin B}{b}=\dfrac{\sin A}{a}\), then \(\sin^{-1}\).
  4. Angle wanted and it could be obtuse? The other answer is \(180^\circ-\) your answer.

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

a and A are a pair: sine rule.

  1. \(\dfrac{b}{\sin65^\circ}=\dfrac{8}{\sin40^\circ}\)
  2. \(b=\dfrac{8\sin65^\circ}{\sin40^\circ}=11.28\)

Answer 11.3 cm

Example \(a=7\) cm, \(A=50^\circ\), \(b=8\) cm. Find angle B, which is obtuse.

Pair a, A: sine rule for an angle.

  1. \(\sin B=\dfrac{8\sin50^\circ}{7}=0.8755\)
  2. calculator: \(61.1^\circ\)
  3. obtuse: \(180^\circ-61.1^\circ\)

Answer 118.9°

The ambiguous case. The calculator only gives the acute angle. If the triangle could also have the obtuse angle \(180^\circ-\theta\) (it still fits with the other angle: \(118.9^\circ+50^\circ<180^\circ\)), there are TWO possible triangles. The question tells you which one ("B is obtuse", or a diagram).

Try it yourself

1
\(a=10\) cm, \(A=100^\circ\), \(B=30^\circ\). Find b (2 d.p.).
Show answer

Answer 5.08

2
\(a=5\), \(A=40^\circ\), \(b=6\). B is obtuse. Find B (1 d.p.).
Hint
sin B = 6 sin 40° ÷ 5, then 180 − that.
Show answer

Answer 129.5

The cosine rule

General methodHow to use the cosine rule

  1. Side wanted: the angle must be BETWEEN the two known sides.
  2. \(a^{2}=b^{2}+c^{2}-2bc\cos A\); work out the right side, then square root.
  3. Angle wanted (three sides): \(\cos A=\dfrac{b^{2}+c^{2}-a^{2}}{2bc}\), where a is opposite the angle.
  4. A negative cosine means the angle is obtuse: the calculator gets it right, no extra step.

Example \(b=7\) cm, \(c=9\) cm and the angle between them \(A=48^\circ\). Find a.

Two sides and the angle between: cosine rule.

  1. \(a^{2}=49+81-126\cos48^\circ=45.69\)
  2. \(a=6.76\)

Answer 6.76 cm

Example A triangle has sides 5 cm, 7 cm and 10 cm. Find its largest angle.

Largest angle is opposite the longest side (10).

  1. \(\cos C=\dfrac{25+49-100}{2(5)(7)}=-0.3714\)
  2. \(C=111.8^\circ\)

Answer 111.8°

Try it yourself

1
Sides 5 cm, 7 cm and 8 cm. Angle opposite the 5 cm side (1 d.p.)?
Show answer

Answer 38.2

2
\(b=5\), \(c=8\), \(A=120^\circ\). Find a (2 d.p.).
Hint
\(\cos120^\circ\) is negative, so you ADD.
Show answer

Answer 11.36

Ways they test it

Type 1 Side with the sine rule

"Calculate AC." (3 marks)

Spot it a side and its opposite angle are both known.

  1. Pair up.
  2. Multiply across.

Type 2 Side with the cosine rule

"Calculate BC." (3 marks)

Spot it two sides and the angle between them.

  1. \(a^{2}=b^{2}+c^{2}-2bc\cos A\).
  2. Square root.

Type 3 Angle from three sides

"Calculate angle PQR." (3 marks)

Spot it all three sides, no angles.

  1. Cosine rule rearranged.
  2. Side opposite the angle goes last with the minus.

For example Sides 5, 7, 10

  1. \(\cos C=-26/70\)

Answer 111.8°

Type 4 Obtuse angle by sine rule

"Calculate the obtuse angle x." (3 marks)

Spot it 'obtuse' with a sine-rule pair.

  1. \(\sin^{-1}\) gives the acute angle.
  2. 180° − it.

Type 5 Show that (1 d.p.)

"Show that angle ACB = 62.9°, correct to 1 d.p." (3 marks)

Spot it 'show that'.

  1. Full working.
  2. Answer to 2 d.p. or more (62.93…) before rounding.

Mistakes to avoid

  • ✗ Cosine rule with an angle that is not between the two sides✓ the angle must be between them, or use the sine rule
  • ✗ \(a^{2}=(b^{2}+c^{2}-2bc)\cos A\)✓ work out \(2bc\cos A\) first, then subtractUse the calculator in one go with brackets right.
  • ✗ Taking the acute angle when it is obtuse✓ \(180^\circ-\theta\)Read the question/diagram: is the angle obviously big?
  • ✗ Pairing a side with the wrong angle✓ each side sits OPPOSITE its angle

Summary

  • Know a side and its opposite angle? Sine rule.
  • Two sides and the angle between, or three sides? Cosine rule.
  • Sine rule angle could be obtuse: \(180^\circ-\theta\).
  • Negative cosine = obtuse angle, automatically.

If you see …

  • a full opposite pairsine rule.
  • SAS or SSScosine rule.
  • 'obtuse'check 180° − θ.

Practice

Practice: easy, medium, hard

Easy

1
You know two sides and the angle between them. Which rule? (sine or cosine)
Show answer

Answer cosine

2
\(\sin x=0.5\). One answer is 30°. The obtuse answer?
Show answer

Answer 150

3
\(A=40^\circ\), \(B=65^\circ\), \(a=8\). Find b (2 d.p.).
Show answer

Answer 11.28

Medium

1
\(b=7\), \(c=9\), \(A=48^\circ\). Find a (2 d.p.).
Guide me

What is it asking? a side

What do we know? SAS: cosine rule

Plan \(49+81-126\cos48^\circ\)

Show working
  1. 45.69
  2. 6.76

Answer 6.76

2
Sides 5, 7, 10. Largest angle (1 d.p.)?
Guide me

What is it asking? an angle

What do we know? SSS: cosine rule

Plan \(\cos C=(25+49-100)/70\)

Show working
  1. 111.8

Answer 111.8

3
\(a=10\), \(A=100^\circ\), \(B=30^\circ\). Find b (2 d.p.).
Guide me

What is it asking? a side

What do we know? pair a, A

Plan \(10\sin30^\circ\div\sin100^\circ\)

Show working
  1. 5.08

Answer 5.08

Hard

1
\(a=7\), \(A=50^\circ\), \(b=8\). B is obtuse. Find B (1 d.p.).
Guide me

What is it asking? an angle

What do we know? \(\sin B=8\sin50^\circ/7\)

Plan acute 61.1°; obtuse = 180 − 61.1

Show working
  1. 118.9

Answer 118.9

PartAnswer / workingMarksGuidance
\(\frac{\sin B}{8}=\frac{\sin50}{7}\)M1
61.1A1
118.9B1
Total: 3 marks
2
\(b=5\), \(c=8\), \(A=120^\circ\). Find a (2 d.p.).
Guide me

What is it asking? a side

What do we know? cosine rule

Plan \(25+64-80\cos120^\circ=129\)

Show working
  1. \(\sqrt{129}\)

Answer 11.36

PartAnswer / workingMarksGuidance
\(25+64-80\cos120\)M1
11.36A1
Total: 2 marks
3
Sides 5 cm, 7 cm, 8 cm. Angle opposite 5 cm (1 d.p.)?
Guide me

What is it asking? an angle

What do we know? SSS

Plan \(\cos=(49+64-25)/112\)

Show working
  1. 38.2

Answer 38.2

PartAnswer / workingMarksGuidance
\(\frac{49+64-25}{2(7)(8)}\)M2
38.2A1
Total: 3 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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