3.10 Applications of the cosine rule

In the last chapter we learnt that the cosine rule can be used with any triangle to help us find:

• the third side of a triangle when you know two sides and the angle between them.

• the angles of a triangle when you know all three sides.

Remember that if $a$a, $b$b and $c$c are the side lengths of a triangle, with angle $C$C opposite the side with length $c$c, then:

The patterns we study are universal, and knowing them well allows us to model and answer questions about the natural world, to explain and measure physical characteristics in a broad range of contexts.

Exploration

Scientists can use a set of footprints to calculate an animal's step angle, which is a measure of walking efficiency.  The closer the step angle is to $180^\circ$180°, the more efficiently the animal walked. Knowing the step angle can help understand how an animal moved, even if they have never seen it. For example, here is a set of prints left by a dinosaur in the lower cretaceous period, with the step angle highlighted: 

This particular set of footprints has lengths $c=304$c=304 cm, $a=150$a=150 cm and $b=182$b=182 cm. We want to find the step angle $C$C to the nearest degree.

We know three side lengths, and want to find an angle, so we can use the cosine rule.

Since angle $C$C is the step angle we're trying to find then we can use this version of the cosine rule:

$\cos C=\frac{a^2+b^2-c^2}{2\times a\times b}$cosC=a2+b2−c22×a×b​

Then substitute in all the values we know:

Notice that the cosine ratio of the angle is negative. This indicates that the angle will be greater than $90^\circ$90°.

Example 1

Example 2