# 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:

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### 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:

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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.

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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:

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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:

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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.