The Right-Angled Triangle

With 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!

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

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!