The Cosine Rule

The cosine rule handles the triangles the sine rule cannot: two sides with the angle between them, or all three sides. Choosing the right rule at each step of a multi-step problem is what this topic really tests.

Proof of the cosine rule

In $△ A B C$ , draw $C D ⊥ A B$ with $C D = h$ and $A D = x$ , so $D B = c - x$ .

In $△ A D C$ : $h^{2} = b^{2} - x^{2}$ and $x = b cos A$
In $△ B D C$ : $a^{2} = h^{2} + (c - x)^{2}$

Substitute:

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

Finding sides and angles

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

Finding a side. $b = 8$ , $c = 11$ , $\hat{A} = 52°$ :

$a^{2} = 64 + 121 - 176 cos 52° \Rightarrow a \approx 8, 75$

Finding an angle. Sides 5, 7 and 9. The smallest angle is opposite the shortest side:

$cos θ = \frac{7 ^{2} + 9 ^{2} - 5 ^{2}}{2 ( 7 ) ( 9 )} = \frac{105}{126} \Rightarrow θ \approx 33, 56°$

A negative cosine means an obtuse angle. The cosine rule never gives an ambiguous case.

Choosing the sine or cosine rule

A side and its opposite angle are known: sine rule.
Two sides and the included angle: cosine rule for the third side.
Three sides: cosine rule for an angle.

In a multi-step problem, it is common to use the cosine rule first and then the sine rule, because once you know a side and its opposite angle, the sine rule is quicker.

Example. Two boats leave a harbour $H$ . Boat $A$ travels 30 km and boat $B$ travels 45 km, with an angle of 115° between their paths. Use the cosine rule for $A B \approx 63, 77$ km, then the sine rule for $\hat{H A B} \approx 39, 76°$ .

Key terms

Cosine rule: $a^{2} = b^{2} + c^{2} - 2 b c cos A$ .
Included angle: The angle between two given sides.
Obtuse angle: An angle between 90° and 180°; its cosine is negative.

Common mistakes

Using the cosine rule with an angle that is not between the two sides.
Calculating $b^{2} + c^{2} - 2 b c$ first and then multiplying by $cos A$ .
Taking the square root of only part of the expression.
Choosing the sine rule for an angle when the cosine rule would avoid the ambiguous case.

Practice questions

Reveal steps one at a time, then tick the marks you earned.

Question 1

4 marks total

routine

In $△ A B C$ , $b = 8$ cm, $c = 11$ cm and $\hat{A} = 52°$ . Calculate $a$ .

Worked solution

Hidden - try the question first

Solution

$a^{2} = 8^{2} + 1 1^{2} - 2 (8) (11) cos 52°$
$a^{2} \approx 76, 64$
$a \approx 8, 75$ cm

Question 2

5 marks total

routine

A triangle has sides 5 cm, 7 cm and 9 cm. Calculate the size of the smallest angle.

Worked solution

Hidden - try the question first

Solution

The smallest angle is opposite the 5 cm side
$cos θ = \frac{7 ^{2} + 9 ^{2} - 5 ^{2}}{2 ( 7 ) ( 9 )}$
$= \frac{105}{126}$
$θ \approx 33, 56°$

Question 3

6 marks total

complex

Prove the cosine rule $a^{2} = b^{2} + c^{2} - 2 b c cos A$ for an acute-angled triangle $A B C$ .

Worked solution

Hidden - try the question first

Solution

Construction: $C D ⊥ A B$ , $C D = h$ , $A D = x$ , $D B = c - x$
In $△ A D C$ : $h^{2} = b^{2} - x^{2}$ and $x = b cos A$
In $△ B D C$ : $a^{2} = h^{2} + (c - x)^{2}$
$a^{2} = b^{2} - x^{2} + c^{2} - 2 c x + x^{2}$
$a^{2} = b^{2} + c^{2} - 2 b c cos A$

Question 4

9 marks total

problem-solving

Two boats leave harbour $H$ at the same time. Boat $A$ travels 30 km and boat $B$ travels 45 km in straight lines. The angle between their paths is 115°.

(a) Calculate the distance between the boats. (3)

(b) Calculate $\hat{H A B}$ . (3)

Worked solution

Hidden - try the question first

Solution

(a) $C D ⊥ A B$
$A B^{2} \approx 4066, 07$
$A B \approx 63, 77$ km
(b) $\frac{sin H A B ^}{45} = \frac{sin 115°}{63 , 77}$
$sin \hat{H A B} \approx 0, 640$
$\hat{H A B} \approx 39, 76°$ (acute, because $115°$ is already obtuse)
(c) In (a) two sides and the included angle were known
so only the cosine rule applies
In (b) side $A B$ and its opposite angle were known, so the sine rule was quicker

FAQs

When do I use the cosine rule?

When you know two sides and the angle between them, or all three sides.

Can the cosine rule give an obtuse angle?

Yes. If the cosine is negative, the angle is obtuse, and the calculator gives it directly.

Is the proof of the cosine rule examinable?

Yes, for acute-angled triangles. The IEB lists it with the Grade 11 content, and the Grade 12 final papers can include Grade 11 work, so keep it ready for both years. Learn the construction and the two uses of Pythagoras.

Proof of the cosine rule

Finding sides and angles

Choosing the sine or cosine rule

Key terms

Common mistakes

Practice questions

FAQs

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