EASIER THAN YOU THINK...

# Trigonometry and Non-Right-Angled Triangles

Although trigonometric ratios were first defined for right-angled triangles (remember SOHCAHTOA?), it is very obvious that most triangles that could be constructed for navigational or surveying reasons would not contain a right angle.  It is also quite obvious that EVERY triangle of any shape or size can be divided into two right-angled triangles.

It was inevitable, therefore, that mathematicians would combine two right triangles in order to create formulae that would apply to ANY triangle.

This led, in particular, to the Sine Rule, the Cosine Rule and the Area Rule.  Sometimes the first two are referred to as the Law of Sine, and the Law of Cosines.

Many students struggle to remember these three formulae.  Fortunately, they are all derived from the same diagram and two of the derivations are very straight-forward!  The first three videos below will explain how to derive each of the three formulae and how to memorise them.  I recommend that you practise deriving them in this way until you can do so from memory.  This is a very good way to learn the formulae and to understand where they come from.

How to Learn and Use The Sine Rule

I strongly recommend that you practise deriving the Sine Rule as part of your plan/strategy to memorise it.

The derivation involves constructing an altitude of a triangle (to create two right-angled triangles), writing out trigonometric ratios for each small triangle, and then solving equations simultaneously.  These are all skills worthy of attention.

Performing this derivation is an exercise that should take you less than two minutes per day.  This is very effective study!

When studying triangles that do NOT contain a right angle, there are three trigonometric formulae that we should know: the Sine Rule, the Cosine Rule, and the Area Rule. Interestingly, they all derive from the same diagram.

In this video I explain how to derive and learn the Sine Rule. At a later date I will produce at least one video explaining how to recognise when to use the rule ... and how to use it.

How to Learn and Use The Area Rule

This very neat little formula allows us to calculate the area of a triangle when we know the length of two sides and the measure of the included angle. In other words, we do not need to know the height, or altitude, of the triangle!

In this video I show how to derive the Area Rule and then explain how to learn it and master its use quickly.

When studying triangles that do NOT contain a right angle, there are three trigonometric formulae that we should know: the Sine Rule, the Cosine Rule, and the Area Rule. Interestingly, they all derive from the same diagram.

In this video I explain how to derive and learn the Area Rule. At a later date I will produce at least one video explaining how to recognise when to use the rule ... and how to use it.

How to Learn and Use The Cosine Rule

Pythagoras' Theorem is a wonderfully useful theorem!  It allows us to calculate the length of a 'missing' side in a right-angled triangle.

The Cosine Rule is, quite simply, Pythagoras' Theorem modified so that it will apply to ANY triangle.

This means that it is our tool of choice whenever we need to calculate the length of a side in any triangle (when given two other sides and an angle*). It even allows us to calculate the size of any angle in a triangle if we know the length of each side!

In this video I show how to derive the Cosine Rule and then explain how to learn it and master its use quickly.

* There is a slight ambiguity here that I will address in a later video.

When studying triangles that do NOT contain a right angle, there are three trigonometric formulae that we should know: the Sine Rule, the Cosine Rule, and the Area Rule. Interestingly, they all derive from the same diagram.

In this video I explain how to derive and learn the Cosine Rule. You will recognise that the Cosine Rule is Pythagoras' Theorem modified for use with ANY triangle!

At a later date I will produce at least one video explaining how to recognise when to use the rule ... and how to use it.

Simply splendid. My concept cleared after 40 years.
Parvez T (on a CCM YouTube video explaining ‘What is an Asymptote?’)

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