And the Cosine Rule: Its easier than it seems

The cosine rule has 4 variables or unknowns:

side a, side b, side c and an angle.

$c^2=a^2+b^2-2ab\cos C$

Or, to find an angle:

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

This means that if you are given 3 of those unknowns, you need to use the cosine rule.

E.g. if you are given 3 sides of a triangle (remember it should not be a right angled triangle.)

Use the cosine rule to find the angle.

Or if you are given 2 sides and the angle IN BETWEEN THOSE 2 SIDES then use the cosine rule to find the 3rd side.

Question 1: Finding a Missing Side

A triangle has sides of length 8 cm and 11 cm, with an included angle of 120°. Calculate the length of the third side to 3 significant figures.

Solution:

We use the cosine rule:

$$c^2=a^2+b^2-2ab\cos C$$

Substituting values:

$$c^2=8^2+{11}^2-2(8)(11)\cos {120}^{\circ }$$

$$c^2=64+121-2(8)(11)(-0.5)$$

$$c^2=64+121+88=273$$

$$c=\sqrt{273}=16.5\text{ cm (to 3 significant figures)}$$

Final Answer: 16.5 cm

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Question 2: Finding an Angle

A triangle has sides of length 10 cm, 12 cm, and 15 cm. Find the largest angle in the triangle to 1 decimal place.

Solution:

The largest angle is opposite the longest side, so we find angle A using the cosine rule:

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

Substituting values:

$$\cos A=\frac{{10}^2+{12}^2-{15}^2}{2(10)(12)}$$

$$\cos A=\frac{100+144-225}{240}$$

$$\cos A=\frac{19}{240}$$

$$A={\cos }^{-1}\left(\frac{19}{240}\right)$$

$$A={85.5}^{\circ }$$

Final Answer: 85.5°

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Question 3: Word Problem – Bearings

Two boats, A and B, leave a port at the same time.

• Boat A travels 20 km due east.
• Boat B travels 25 km on a bearing of 40° from the port.

Find the distance between the two boats to 1 decimal place.

Solution:

We form a triangle where:

• $a=20$ km
• $b=25$ km
• Included angle $C={40}^{\circ }$

Using the cosine rule:

$$c^2=a^2+b^2-2ab\cos C$$

Substituting values:

$$c^2={20}^2+{25}^2-2(20)(25)\cos {40}^{\circ }$$

$$c^2=400+625-2(20)(25)(0.766)$$

$$c^2=1025-766=259$$

$$c=\sqrt{259}=16.1\text{ km}$$

Final Answer: 16.1 km

Cosine Rule Triangle Calculator - beta mode

Enter two sides and the included angle (in degrees):

Side a: Side b: Angle C (°): Calculate

More examples and practice questions coming soon!