Music from the Triangle

You too can make sweet music by simply using a right-angled triangle and a bit of "trig".

Trigonometry is the technique you use to find unknown angles or lengths of sides of right-angled triangles.

Anyone who needs to measure and calculate angles and lengths of objects in their work will always encounter right angled triangles (e.g. house roofs, bridge supports, garden plots, road slope, banking of rally car in a computer game...etc.). So, trig is useful for engineers, designers, architects, computer games programmers and a host of other people.

There are 3 basic trigonometric functions which you should become familiar with.

All of the functions are related to the two angles in the triangle which are not 90 degrees


The 3 functions are called tangent, sine and cosine.
The 3 functions are commonly shortened to tan, sin and cos (not to be confused with the results of a day in the sun, naughty behaviour, and a wierd lettuce).

If we use the triangle above, we can define what each of these functions mean.

tan a   =  Opposite / Adjacent

sin a   =  Opposite / Hypotenuse

cos a  =  Adjacent / Hypotenuse

In words we are saying for example that: the tangent of the angle a (measured in degrees) is equal to the length of the side of the triangle which is opposite to angle a divided by the length of the side of the triangle which is adjacent to (next to) angle a.

Note that you could just have easily named the other angle in the triangle a, and the formulas will still apply (provided you switch the opposite and adjacent labels so that they relate to the new angle).

About 20 years ago before calculators came on the scene, there used to be a profitable industry for publishers who would create mathematical "trig" tables which made great bedtime reading (not). Luckily now this trig stuff is much easier to do. All you need to know are the formulas above and you can get your calculator to do the real work for you!

Here are some trig examples where you need to find the unknown x:

From the formula we know that: sin x = opposite / hypotenuse
hence we can say that sin x = 5/8.

Now all we need to do is find x - simply use the sin-1 (inverse sine) function on your calculator the inverse sine of 5/8 turns out to be 38.68 (to 2 decimal places) hence the angle x = 38.68 degrees.

From the formula we know that tan x = opposite / adjacent
hence we can say that tan x = 4/6.

Using the calculator again (this time using the inverse tangent button) we can easily say that the angle x = 33.69 degrees. (to 2 d.p.)

From the formula we know that cos x = adjacent / hypotenuse
hence we can say that cos x = 3/4.

Using the calculator again (this time using the inverse cosine button) we can easily say that the angle x = 41.41 degrees (to 2 d.p.)

Here you need to find the length of a side rather than an angle - you can still use the same formulas. Here sin 30 = x/15. If we do a little algebra we can rearrange this to 15 sin 30 = x.

Find sin 30 on your calculator (which gives the answer 0.5) then multiply by 15 to get the length of side x. So side x = 7.5m.


Many people find it tricky to learn these formulas, there are various ways of doing this. One is to remember the nonsense word SOHCAHTOA (which can be tricky).

If you know how to calculate gradients, you can think of tan as the gradient of a tangent (line touching the edge of a circle).

sin oh (as in "oh no I've sinned")

cos ah (as in "ah what a tasty lettuce") and so on...

If you try to think of your own way to remember these formulas, by the time your half way through - I bet you'll remember them naturally anyway. Try it and see!