The Cosine Rule

Use the cosine rule to find a side from two sides and the angle between them, find an angle from three sides, and set up an equation when the sides are in algebra.

In the exam for Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024, Edexcel Further Pure Maths 4PM1, Edexcel IGCSE Maths 4MA1, IGCSE Additional Maths 0606 · 66 real past-paper questions below

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The Cosine Rule

Use it when a triangle has no right angle and no side with its facing angle known: two sides and the angle between them, or all three sides.

Formula

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

$A$ is the angle between sides $b$ and $c$ . Side $a$ is the one facing it, across the triangle.

Key idea

If $A = 9 0^{\circ}$ , then $cos A = 0$ and the rule becomes Pythagoras. The last part corrects for the angle not being 90°.

Key idea

In France, the cosine rule is called théorème d'Al-Kashi. It is named after Jamshid al-Kashi, a Persian mathematician in Samarkand who wrote it out as a clear rule in 1427.

Finding a Side

The 3-mark question: two sides and the angle between them. Find the third side.

Worked example

1
$b = 7, c = 10, A = 3 8^{\circ}$ — Label the triangle. The 38° is between the 7 and the 10, so those are b and c. The side we want, a, is across from the angle.
2
$a^{2} = b^{2} + c^{2} - 2 b c cos A$ — Write the rule first.
3
$a^{2} = 7^{2} + 1 0^{2} - 2 (7) (10) cos 3 8^{\circ}$ — Put the numbers in.
4
$7^{2} + 1 0^{2} = 49 + 100 = 149$ — Square each side and add them.
5
$2 (7) (10) cos 3 8^{\circ} = 140 \times 0.7880 = 110.32$ — Multiply the whole last part together. Do this before any subtracting.
6
$a^{2} = 149 - 110.32 = 38.68$ — Now subtract. This is a², not a yet.
7
$a = 38.68 = 6.22 cm$ — Take the square root to get a, and keep the positive one because a length cannot be negative.

Key idea

For a “show that” question the answer is given, such as a = 6.22. Write a² with more decimal places, such as 38.68…, then the rounded answer. Keep full calculator values until the end.

Common mistake

Working out $149 - 140$ first and then multiplying by cos. Work out $140 cos 3 8^{\circ}$ first, then subtract.

When the Angle Is Obtuse

If the angle between the two sides is more than 90°, its cosine is negative. Then the minus in the rule meets another minus. This is where marks are lost.

Worked example

1
$b = 9, c = 12, A = 12 5^{\circ}$ — The 125° is between the 9 and the 12, so a is the side across from it.
2
$a^{2} = 9^{2} + 1 2^{2} - 2 (9) (12) cos 12 5^{\circ}$ — Write the rule with the numbers in.
3
$cos 12 5^{\circ} = - 0.5736$ — An angle over 90° has a negative cosine. The calculator shows the minus.
4
$a^{2} = 225 - 216 \times (- 0.5736)$ — 81 + 144 = 225 and 2 × 9 × 12 = 216. Put the negative number in brackets so its sign is not lost.
5
$a^{2} = 225 + 123.9 = 348.9$ — A minus times a minus is a plus, so the 123.9 is added, not subtracted.
6
$a = 348.9 = 18.7 cm$ — Take the square root and give 3 significant figures.
7
$18.7 > 12 > 9$ — Check it. The side across from the biggest angle must be the longest, and it is.

Common mistake

Subtracting the 123.9. That gives 10.1 cm, which is shorter than the 12 cm side. A side across from an obtuse angle is always the longest, so the answer must be bigger.

Finding an Angle From Three Sides

The 3-mark question gives all three sides. Rearrange the rule so the cosine is on its own.

Formula

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

Sides of 6 cm, 9 cm and 13 cm. Find the largest angle. It faces the longest side, so $a = 13$ .

Worked example

1
$a = 13, b = 6, c = 9$ — The largest angle faces the longest side, so the 13 is a. The other two sides are b and c.
2
$cos A = \frac{6 ^{2} + 9 ^{2} - 1 3 ^{2}}{2 \times 6 \times 9}$ — Put the sides into the rearranged rule. The squared side a goes last, with the minus.
3
$cos A = \frac{36 + 81 - 169}{108}$ — Square each side, and 2 × 6 × 9 = 108.
4
$cos A = \frac{- 52}{108}$ — 36 + 81 = 117, and 117 − 169 = −52. A negative top is fine.
5
$cos A = - 0.4815$ — Divide.
6
$A = cos^{- 1} (- 0.4815) = 118. 8^{\circ}$ — Press inverse cosine. A negative cosine gives an angle over 90°, which is right for the largest angle.

Key idea

Found the largest angle? Find the next one with the sine rule, then the third by taking both away from 180°. Only the largest angle can be obtuse, so the sine rule gives just one answer.

Key idea

A negative cosine means the angle is obtuse, over 90°. The calculator handles it for you.

When the Sides Are in Algebra

The 5-mark question: the sides are expressions and the angle is 60°. Use $cos 6 0^{\circ} = \frac{1}{2}$ to make an equation.

Sides $x$ and $x + 7$ have 60° between them. The side facing it is 13 cm. Find the perimeter.

Worked example

1
$1 3^{2} = x^{2} + (x + 7)^{2} - 2 (x) (x + 7) cos 6 0^{\circ}$ — The 60° is between x and x + 7, and 13 is across from it, so 13 plays the part of a.
2
$1 3^{2} = x^{2} + (x + 7)^{2} - 2 (x) (x + 7) \times \frac{1}{2}$ — cos 60° is exactly ½, so there are no decimals.
3
$169 = x^{2} + (x + 7)^{2} - x (x + 7)$ — 13² = 169, and 2 × ½ = 1.
4
$169 = x^{2} + (x^{2} + 14 x + 49) - (x^{2} + 7 x)$ — Expand each bracket. The minus in front of the last one changes both of its signs.
5
$169 = x^{2} + 7 x + 49$ — Collect like terms: x² + x² − x² = x², and 14x − 7x = 7x.
6
$x^{2} + 7 x - 120 = 0$ — Take 169 from both sides, so 49 − 169 = −120.
7
$(x - 8) (x + 15) = 0$ — Factorise: the numbers multiply to −120 and add to 7, so −8 and 15.
8
$x = 8 or x = - 15$ — Each bracket can be zero.
9
$x = 8$ — Reject −15. A length cannot be negative.
10
$8 + (8 + 7) + 13 = 36 cm$ — The perimeter is all three sides: x = 8, x + 7 = 15 and the 13.

Common mistake

Stopping at $x = 8$ . The question asks for the perimeter, so add all three sides.

Which Method? Read the Question

Look for a side and the angle facing it.

You see	You do
Two sides and the angle between	cosine rule for the third side
Three sides, find an angle	cosine rule rearranged
Largest angle	it faces the longest side
A side and the angle facing it	The Sine Rule
More than one step	Choosing Between Sine and Cosine Rules

Key idea

Label the angle you use $A$ and the side across from it $a$ first. Then the rule cannot go wrong.

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