Getting to Grips with Trigonometry

Age 16 to 18
Article by Alison Kiddle

Published 2013 Revised 2016

When sitting STEP and other advanced mathematics examinations, you are almost certain to come across questions involving trigonometry. To maximise your chances of performing well on these questions, we have put together a few problems to develop your understanding of trig.

When you met Sine, Cosine and Tangent for the first time, it was probably in the context of finding missing sides and angles of right-angled triangles, but there are other useful ways of thinking about the trigonometric functions. The problem Trig Reps introduces some of these, and challenges you to think about the advantages of a certain represenation for different purposes. In order to succeed on STEP and other advanced mathematics examinations you will need to be familiar with thinking about trigonometric functions as functions, which can be represented by a series expansion, and if you are sitting STEP III it will be useful to have some familiarity with the complex representation of the trig functions. 

One of the lovely properties of the trig functions is the many relationships between them. If you have never seen it before, I recommend taking a look at the animation in the NRICH problem Sine and Cosine, which uses a unit circle to show relationships between the sine and cosine of an angle. Think about how the sine and cosine change with the angle, and why this gives rise to the familiar oscillating curve of the graphs $y=\sin x$ and $y= \cos x$. Also, consider why one graph is a translation of the other. The well-known trig identity $\sin^2 x + \cos^2 x = 1$ becomes obvious if you define Sine and Cosine in terms of the lengths on the unit circle and you are familiar with Pythagoras's Theorem!

Once you are happy with different models for thinking about the trig functions, familiarise yourself with the common relationships - double angle formulae, addition formulae and so on. Even those formulae that appear in the formula book should be at your fingertips and you should be confident at applying them accurately under examination conditions.

Another at-your-fingertips skill needs to be integrating and differentiating the trig functions, and transforming their graphs. Can you relate your understanding of the graphs to the integrals and derivatives? You might wish to use GeoGebra to plot some graphs of trig functions and consider how transformations of functions affect the derivatives and integrals.

Finally, remember some very basic properties of trig functions: $\sin x$ and $\cos x$ always have a value between $-1$ and $1$, and it's very helpful to remember (in radians, of course) where each function has key values such as $\frac{1}{2}$, $\frac{\sqrt{3}}{2}$, $\frac{1}{\sqrt{2}}$ and of course $0, 1$ and $-1$.

Why not practise some trig questions using our Interactive Workout?