Sine & Cosine Rules (CIE IGCSE Maths: Extended)

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17 November 2022

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The sine rule allows us to find missing side lengths or angles in non-right-angled triangles
It states that for any triangle with angles A, B and C

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

- Where
  - a is the side opposite angle A
  - b is the side opposite angle B
  - c is the side opposite angle C
Sin 90° = 1 so if one of the angles is 90° this becomes SOH from SOHCAHTOA

Non-Right-Angled Triangles Diagram 1a

The sine rule can be used when you have any opposite pairs of sides and angles
Always start by labelling your triangle with the angles and sides
- Remember the sides with the lower-case letters are opposite the angles with the equivalent upper-case letters
Use the formula to find the length of a side
To find a missing angle you can rearrange the formula and use the form

$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$

The following diagram shows triangle ABC. $AB = 8.1 cm$ , $BC = 12.3 cm$ , angle $B C A = 27 °$ .

3-3-2-sine-rule-we-question

Use the sine rule to calculate the value of:

x

3-3-2-ai-sl-sine-rule-we-solution-i

ii)

y

3-3-2-ai-sl-sine-rule-we-solution-ii

The cosine rule allows us to find missing side lengths or angles in non-right-angled triangles
It states that for any triangle

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

- Where
  - a is the side opposite angle A
  - b and c are the other two sides
Both of these formulae are rearrangements of the same thing
- The first version is used to find a missing side
- The second version is a rearrangement of this and can be used to find a missing angle
Cos 90° = 0 so if A = 90° this becomes Pythagoras’ Theorem

The cosine rule can be used when you have two sides and the angle between them or all three sides
Always start by labelling your triangle with the angles and sides
- Remember the sides with the lower-case letters are opposite the angles with the equivalent upper-case letters
Use the formula $a^{2} = b^{2} + c^{2} - 2 b c \cos A$ to find an unknown side
Use the formula $\cos A = \frac{b^{2} + c^{2} - a^{2}}{2 b c}$ to find an unknown angle
- A is the angle between sides b and c
Substitute the values you have into the formula and solve