Trigonometry

Trigonometry is one of those names you always hear about in mathematics.  The topic of trigonometry starts off being about right-angled triangles.  It’s actually all about ratios between the side lengths in right-angled triangles.  Let’s start with a typical right-angled triangle:

Naming the sides

Now, first up, there are three different names for the three different sides of a right-angled triangle.  The names are opposite, hypotenuse and adjacent.  Now we’ve come across the hypotenuse before – it is the longest side of the triangle.  This is always easy to spot.  The other two names, opposite and adjacent, depend on which angle you’re currently looking at in the triangle.  For instance, say I was looking at angle A:

The opposite side is the side opposite the angle we’re looking at.  We’re looking at angle A at the moment, so the side opposite it is the side on the left:

This leaves us with the adjacent side.  The adjacent side is the side of the triangle that touches the angle we’re looking at, but which is not the hypotenuse.  There are two sides touching our angle A – one is the hypotenuse.  The other side therefore is the adjacent side:

What about if we’d picked another angle, say angle B in the following diagram?  Well, the hypotenuse would stay the same, but the adjacent and opposite sides would change, like this:

We don’t usually have to worry about how to name the sides when the angle we’re looking at is the 90 degree angle, so don’t worry about that for the moment.

Ratios between the side lengths

Let’s go back to the ‘A’ angle triangle:

Trigonometry is all about the ratio of the side lengths in the triangle.  For instance, when we’re looking at the angle A, we could talk about the ratio between the length of the adjacent side and the length of the hypotenuse:

                                     

Now when we talk about this ratio, we have to remember what angle we’re currently looking at in the diagram – angle A.  There is a special name in trigonometry for this ratio we have just looked at – it is known as ‘cosine A’.  When we say ‘cosine A’, what we mean is the ratio between the length of the adjacent side and the hypotenuse side.  Often we use ‘cos’ instead of ‘cosine’ as a shorter name.

                            

There are two other ratios you need to know about.  The first is ‘tangent A’ – it is the ratio between the length of the opposite side and the adjacent side.  We use ‘tan’ for short.  The other is ‘sine A’ – it is the ratio between the length of the opposite side and the hypotenuse.  We use just ‘sin’ for short.  Here’s a little summary of the three ratios:

                           

Now, there’s an easy way to remember what all these ratios are – SOH CAH TOA.  Say it out aloud – it is a word you can say easily and should be able to remember after saying it a few times.  The way to use it is to look at each of the letters in it, which stand for the following:

                                  SOH = (S)in : (O)pposite over (H)ypotenuse

                                CAH =  (C)os : (A)djacent over (H)ypotenuse

                                   TOA = (T)an : (O)pposite over (A)djacent

So say I have a triangle like this one, and I’m interested in tangent θ:

First I need to label the names of the sides.  The longest side is the hypotenuse.  The side opposite the angle θ is the opposite side.  The side touching the angle θ which is not the hypotenuse is the adjacent side:

Now, I remember my SOH CAH TOA.  Which part am I interested in?  Well the question is asking for tangent θ, or just tan θ for short.  This means I’m interested in the last part of the word – the ‘TOA’ bit:

                                   TOA = (T)an : (O)pposite over (A)djacent

So the ratio I’m looking for is the length of the opposite side divided by the length of the adjacent side:

                                       

This is the basic procedure you need to follow whenever you need to find cos, sin or tan of an angle.  Look at where the angle is in the triangle, and label the sides of the triangle.  Then, using SOH CAH TOA, work out which ratio you need.