crystal clear mathematics logo





Make a Donation to Crystal Clear Mathematics

Sign up to my Newsletter


The Right-Angled Triangle

Image of Three Right Triangles Showing the Three Trigonometric Functions and SOH-CAH-TOATriangles have three sides.  Geometricians quickly realised that if a triangle remained the same SHAPE (i.e. all three angles remaining the same), then the ratio of sides remained constant.  A triangle composed of 3 m, 4m, and 5m lengths, if tripled in size, would now be 9m, 12 m, and 15 m.  However, the ratios would remain constant … 9/12 = 3/4, 12/15 = 4/5, and 9/15 = 3/5.  These ratios could be related to the fixed angles in such triangles.  Each different triangle, with its different angles, would have its own particular ratios of sides.

In time, mathematicians (after a great deal of clever calculations) drew up tables recording all these ratios, together with the angles that were associated with them.  They were able to use this information to solve many practical and scientific problems.  These were problems relating to reflection and refraction of light, of surveying and navigation, and of calculating the heights of mountains around settlements (many of which still have “trig stations” on top … structures that were used for trigonometry!).


Learning the Main Three Ratios and SOHCAHTOA

Image of Three Right Triangles Showing the Three Trigonometric Functions and SOH-CAH-TOAWith three sides in each triangle, there are six possible ratios, and they have all been given names.






The three "main" ratios are:

  • sine(θ) = opposite/hypotenuse,
  • cosine(θ) = adjacent/hypotenuse, and
  • tangent(θ) = opposite/adjacent

The three "reciprocal" ratios are:

  • cosecant(θ) = hypotenuse/opposite,
  • secant(θ) = hypotenuse/adjacent, and
  • cotangent(θ) = adjacent/opposite

In time, I will explain to you why they have the names that they do.  For the moment, it is sufficient to know about them and to know that we will not use the reciprocal ratios for some time.  This means that you only have to learn the main three ratios at this stage.  To help you do this, there are a couple of basic rules to recognise.

First, to make life easier, mathematicians agreed to simplify the six ratios to the following:

  • sin (which we still pronounce "sine"),
  • cos (which we generally pronounce "coz"),
  • tan (which we pronounce "tan"),
  • cosec (which is written "csc" in the USA and is pronounced "cosec" throughout the world),
  • sec (pronounced "sec") and
  • cot (pronounced "cot).

We even abbreviated the names of the sides:

  • opp(osite),
  • adj(acent) and
  • hyp(otenuse)

although we still speak of them as "opposite," "adjacent," and "hypotenuse" (NOT "opp," "adj," and "hyp").

Second, a memory aid has been devised (and used all over the world) which helps students remember those three main ratios.  Using the first letter of each part of the ratio:

  • Sin(θ) = Opp/Hyp is remembered by SOH,
  • Cos(θ) = Adj/Hyp is remembered by CAH, and
  • Tan(θ) = Opp/Adj.
  • Put together, they form the 'nonsense' word SOH-CAH-TOA.

You may want to print out a copy of the graphic that I created (see above) and place it near your study desk as a reminder.  To learn the ratios properly, however, there are some better things to do than simply stare at a piece of paper!

First, although it does not take long for people to remember the expression SOH-CAH-TOA, they don't always remember how each part is SPELLED!  To overcome this problem, you can do two things.

  • Learn a mnemonic (memory aid) such as Some Old Hags Can Always Hide Their Old Age.  There are many others!  See here for a fascinating collection of songs and acronyms and other ideas!
  • You should also WRITE the mnemonic at every opportunity.  I instruct my students to write SOHCAHTOA at the right of their page as they answer each trigonometric question.  It will not take very long before you will know these three ratios very well!
How to Learn the Exact Ratios

When I was at school, I knew students who tried to memorise exact ratios for all relevant angles up to 360 degrees!

An Illustration of How the Two Triangles Are Created In Oder to Evaluate Exact Ratios

This is a long and difficult exercise and can be so easily avoided if students invest (literally) a few minutes each day in learning the structure of the key triangles.  It is because students have poor learning strategies that so many of them struggle to remember the correct ratios. You will notice that I continue to recommend the same learning system:

  • Set specific time aside to focus and learn quickly without distractions.
  • Practise deriving and using the material intensively for a few days.
  • Immediately use the formula/material to solve some problems.
  • Use your diary to plan further revision BEFORE you forget what you have learned.

I hope this helps!

I don’t know if you remember me, but you really helped me last year and I just wanted to tell you I came first in my maths half yearly with 100%. A lot of it goes to you. Thanks for your help.
Jashan M (student, 2013)

See all Testimonials

Sign up to my Newsletter

Keeping it Social




Make a Donation to Crystal Clear Mathematics

Copyright © Crystal Clear Mathematics | All Rights Reserved

Website Design: | Photography: