9.1PureCore

The Cosine Rule

Right-angled trigonometry only works on right-angled triangles. The cosine rule extends it to ANY triangle, letting you find a missing side from two sides and the angle between them, or a missing angle from all three sides.

30 min Video by Zeeshan Zamurred Trigonometric Ratios

Edexcel AS Level Maths: 9.1 The Cosine RuleWatch the full walkthrough before the notes below.

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What you'll be able to do

State the cosine rule
Find a missing side using two sides and the included angle
Rearrange the rule to find a missing angle
Recognise when the cosine rule (not the sine rule) is needed

The rule

For a triangle with sides $a$ , $b$ , $c$ and the angle $A$ opposite side $a$ , the cosine rule links all three sides to one angle. It is essentially Pythagoras with a correction term for the angle.

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

Side

a

is opposite angle

A

;

b

and

c

enclose that angle.

Tip — Label the triangle so the side you want (or the angle you want) is the “a”/“A” in the formula.

Finding a side

Use this form when you know two sides and the $included$ angle (the angle between them) — often called the SAS case.

1Substitute:

a^{2} = 7^{2} + 5^{2} - 2 (7) (5) cos 6 0^{\circ}

a^{2} = 49 + 25 - 70 (0.5) = 74 - 35 = 39

a = 39

Answer

a \approx 6.24

Finding an angle

Rearranged, the cosine rule finds an angle when you know all three sides (the SSS case).

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

Put the side opposite the wanted angle as

a

cos A = \frac{5 ^{2} + 7 ^{2} - 8 ^{2}}{2 ( 5 ) ( 7 )} = \frac{25 + 49 - 64}{70} = \frac{10}{70}

A = cos^{- 1} (0.1429)

Answer

A \approx 81. 8^{\circ}

Tip — If cos A comes out negative, the angle is obtuse (greater than 90°) — that is fine.

Formula recap

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

Cosine rule — find a side (SAS).

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

Rearranged — find an angle (SSS).

Common mistakes to avoid

Using the cosine rule when you only know one side and its opposite angle.

That is the sine rule’s job. Cosine rule needs SAS (a side) or SSS (an angle).

Pairing the wrong side with the angle.

The side a must be OPPOSITE the angle A in the formula.

Key takeaways

Cosine rule: a² = b² + c² − 2bc·cos A, with a opposite A.
Find a side from two sides + the included angle (SAS).
Rearrange to cos A = (b² + c² − a²)/(2bc) to find an angle (SSS).
A negative cosine signals an obtuse angle.

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