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 $$a, $$b and $$c are the side lengths of a triangle, with angle $$C opposite the side with length $$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°, 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 $$c=304 cm, $$a=150 cm and $$b=182 cm. We want to find the step angle $$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 is the step angle we're trying to find then we can use this version of the cosine rule:

$$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°.

Example 1

Example 2