11.07 Cosine rule

The cosine rule (or the law of cosines) relates the lengths of the sides and the cosine of one of its angles. The cosine rule is useful for finding:

• the third side of a triangle given two known side lengths and the angle between them, or

• the angles of a triangle given all three sides lengths are known.

Consider the triangle \triangle ABC, the cosine rule can be written as any one of the following equations: a^2 = b^2 + c^2 - 2bc\cos A \\ b^2 = a^2 + c^2 - 2ac\cos B \\ c^2 = a^2 + b^2 - 2ab\cos C

Notice that  Pythagoras' theorem  c^2= a^2 + b^2 makes an appearance in the cosine rule c^2 = a^2 + b^2 - 2ab\cos C when C has a measure of 90\degree which makes the term 2ab\cos C equal to zero.

If the three side lengths in a triangle are a, \,b and c, with an angle C opposite the side with length c, then: c^2=a^2+b^2-2ab\cos C

Note: The formula can be written in terms of any of the sides or angles, familiarise yourself with the forms below and note the pattern. If a triangle is labelled differently we can adapt the rule using the pattern.

Examples

Example 1

Find the length of a using the cosine rule. Round your answer to two decimal places.

Worked Solution

Create a strategy

Use the cosine rule a^2=b^2+c^2-2bc\cos A.

Apply the idea

Since we know two sides and one angle, we can use the cosine rule to find a.

\displaystyle a^2	\displaystyle =	\displaystyle b^2+c^2-2bc\cos A	Use the cosine rule
\displaystyle a^2	\displaystyle =	\displaystyle 13^2+30^2-2\times 13\times 30\cos 33\degree	Substitute b=13, \, c=30, \, A=33\degree
\displaystyle a	\displaystyle =	\displaystyle \sqrt{13^2+30^2-2\times 13\times 30\cos 33\degree}	Take the square root of both sides
	\displaystyle \approx	\displaystyle 20.37	Evaluate and round

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Example 2

Find the length of c using the cosine rule. Round your answer to two decimal places.

Worked Solution

Create a strategy

Use the cosine rule c^2=a^2+b^2-2ab\cos C.

Apply the idea

Since we know two sides and one angle, we can use the cosine rule to find c.

\displaystyle c^2	\displaystyle =	\displaystyle a^2+b^2-2ab\cos C	Use the cosine rule
\displaystyle c^2	\displaystyle =	\displaystyle 25^2+15^2-2\times 25\times 15\cos 25\degree 35'	Substitute a=25, \, b=15, \, C=25\degree 35'\degree
\displaystyle c	\displaystyle =	\displaystyle \sqrt{25^2+15^2-2\times 25\times 15\cos 25\degree 35'}	Take the square root of both sides
	\displaystyle \approx	\displaystyle 13.17	Evaluate and round

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When looking to solve for an unknown angle, the equation can also be rearranged and written as: \cos C = \dfrac{a^2 +b^2 - c^2}{2ab}

To find a missing angle using the cosine rule, we need to know all three side lengths.

Examples

Example 3

Find the value of angle W using the cosine rule. Round your answer to two decimal places.

Worked Solution

Create a strategy

Use the cosine rule for angles: \cos C = \dfrac{a^2 +b^2 - c^2}{2ab}.

Apply the idea

Since we know all three side lengths, we can use the cosine rule to find the angle.

\displaystyle \cos C	\displaystyle =	\displaystyle \dfrac{a^2+b^2-c^2}{2ab}	Use the cosine rule
\displaystyle \cos w	\displaystyle =	\displaystyle \frac{32^2+19^2-25^2}{2\times 32\times 19}	Susbtitute a=32, \, b=19, \, c=25,\,C=w
\displaystyle \cos w	\displaystyle =	\displaystyle \dfrac{5}{8}	Evaluate the right side
\displaystyle w	\displaystyle =	\displaystyle \cos ^{-1}\left(\frac{5}{8}\right)	Take the inverse cosine of both sides
	\displaystyle \approx	\displaystyle 51.32\degree	Evaluate and round

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Mathematics is often thought of as being confined to the classroom and being separate from the real world. Students may ask themselves "What's the point of learning the cosine rule?" The truth is, part of the beauty in mathematics is that it can explain and measure so much of what is happening in the real and natural world.

Examples

Example 4

Find the length of the diagonal, x, in parallelogram ABCD. Round your answer to two decimal places.

Worked Solution

Create a strategy

Use the cosine rule c^2=a^2+b^2-2ab\cos C.

Apply the idea

\displaystyle c^2	\displaystyle =	\displaystyle a^2+b^2-2ab\cos C	Use the cosine rule
\displaystyle x^2	\displaystyle =	\displaystyle 18^2+35^2-2\times 18\times 35\cos 130\degree	Substitute c=x, \, a=18, \, c=35,\,C=130\degree
\displaystyle x	\displaystyle =	\displaystyle \sqrt{18^2+35^2-2\times 18\times 35\cos 130\degree}	Take the square root of both sides
	\displaystyle \approx	\displaystyle 48.57	Evaluate and round

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Example 5

Dave leaves town along a road on a bearing of 169\degree and travels 26 \text{ km}. Maria leaves the same town on another road with a bearing of 289\degree and travels 9\text{ km}. Find the distance between them, x. Round your answer to the nearest \text{km}.

Worked Solution

Create a strategy

Sketch the situation to identify the sides and angle.

Apply the idea

Here's the diagram showing the situation:

We have the side lengths, a=x,\,b=26,\,c=9, and one angle, A=289\degree - 169\degree = 120\degree.

\displaystyle a^2	\displaystyle =	\displaystyle b^2+c^2-2ab\cos A	Use the cosine rule
\displaystyle x^2	\displaystyle =	\displaystyle 26^2+9^2-2\times 26\times 9\cos 120\degree	Substitute the values
\displaystyle x	\displaystyle =	\displaystyle \sqrt{26^2+9^2-2\times 26\times 9\cos 120\degree}	Take the square root of both sides
	\displaystyle \approx	\displaystyle 31 \text{ km}	Evaluate and round

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