### Trigonometry: Triangles to Functions

Many ways problem

# Slices of $\pi$ Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

## Things you might have noticed

Take a look at the diagram below.

We’ve used inequalities involving $\sin \theta$, $\cos \theta$ and $\tan \theta$ to divide the semicircle into sectors. Each sector in the diagram is defined by a different inequality. For example, one sector is defined by the angles $\theta$ between $0$ and $\pi$ for which $\cos \theta < \sin \theta < \tan \theta.$ Another sector is defined by $\cos \theta< \tan \theta < \sin \theta.$

• Can you work out which inequality has been used to define each sector?

There are many ways to approach this problem, and we will illustrate a few of them here. If you have used different approaches, or used similar approaches but for different parts of the problem, you might find it interesting to compare the methods.

However, we may need to do a bit more work to complete the sketch graphs. For example, do the graphs $y=\tan \theta$ and $y=\sin \theta$ cross between $\theta = 0$ and $\theta =\dfrac{\pi}{2}$? Here are a couple of ways to think about this.

We now only need to decide which inequality defines the orange sector. You may find it helpful to look at the graphs or to visualise a point moving round the unit circle.

• Which is the biggest sector?

The boundary of the orange and green sectors is at $\theta=\dfrac{\pi}{4}$, and therefore the green sector is larger than the red sector. However, the yellow or blue sectors may be bigger than the green sector. Here are a few ways to tackle this problem.

If you used more than one approach, can you connect them? For example, can you connect the symmetry argument with solving $\tan \theta = \cos \theta$?

• If you extended the diagram to make a complete circle, how many extra sectors would you need?

To complete the circle we can use similar methods to those discussed above. Another approach might be to consider the possible orderings of $\sin \theta$, $\cos \theta$ and $\tan \theta$.

Can you explain how the behaviour of the functions gives us these sectors and inequalities?

There were many ways to approach this problem. Instead of considering all three functions at the same time, we could have considered pairs of functions.

• When is $\sin \theta < \cos \theta$?

• When is $\sin \theta < \tan \theta$?

• When is $\tan \theta < \cos \theta$?

Can you see how these inequalities feature in the diagram above?