# Trigonometry to Begin With

## Trickonemetry Rules !

Having an introduction into** the World of Shapes** in The Shape of Things to Come, let us now have a better look at** Geometry** and its related topics, which themselves form a large part of the study of **Mathematics**.

First, but by no means easiest, I would like to sneak a quick detour to that one subject which floors many hardworking students : **Trigonometry**.

The reason for our short look at this, is that we shall be using some of it for other related topics, so let us now refresh our memories. You may or may not have done this kind of thing before, but once you get the hang of it, it isn’t so hard, and the main thing is understanding what it all *means*.

## What it Rules

**Trigonometry especially involves the study of right angled triangles**, as well as **Circular Functions which are graphed as Waves**, and other kinds of curves, but we won’t concern ourselves with *that* branch of the subject as yet. In any** Triangle**, the internal angles all add up to **180 **degrees, so if one of them is at **ninety**, the sum of the other two angles add up to the other **ninety**.

**The ancient Greeks** **studied Shapes in particular** - as opposed to the **Arabs’ fascination with Numbers and Letters** - and they eventually worked out that, according to the* size * of the other two smaller internal angles, there was a relationship between the lengths of the sides of the **Triangle**. Now these relationships are always the same regardless of the size of the **Triangle**, as long as the angles within the **Triangle** are identical to those within the **Triangle** of the other size, where we say they are **Similar** **Triangles**.

Once again, just so we can see what is being meant, let us draw ourselves a little picture, to make the whole thing that much clearer :

## What the above Diagram means

If you look at the **Triangle**, we concern ourselves with the **angle** at the bottom left, marked by the** Greek** letter **theta **( **θ** ), as is the custom. With respect to this angle only,** the right hand vertical side is its Opposite**, the **bottom edge** ( **base of the** **triangle** ), **is its Adjacent**, while** the longest side is always the Hypotenuse**.

Let us say that **the angle** **θ** is equal to **30°**, so in order for it and the other angle to both add up to **ninety**, then this other angle, denoted here as **delta** ( **δ** ), has to equal **60°**. The idea behind **the three main Trigonometric Functions** ( **Sine**,** Cosine** and **Tangent** ), is that they give us a **ratio of the lengths of the sides** - that is, they tell us how much longer one side is to the other, depending on the size of the angle.

Let us then decide to compare the sides. We shall say that the base of this **Triangle** is **five** inches in length. From just this, we can actually work out the lengths of the other two sides.

The base, in relation to the angle we shall use, **θ**, which is **30°**, is its **Adjacent side**, so we need to use a function involving the **Adjacent**. Our three main functions are :

**Sine = Opposite divided by Hypotenuse**

**Cosine = Adjacent divided by Hypotenuse**

**Tangent = Opposite divided by Adjacent**

The choice of function is determined by which length you want to find. Say I would like to know how long the** Hypotenuse** is. Since I already know the **Adjacent**, which equals **five** inches, the obvious one to choose is **the Cosine**, as this will tell me what decimal fraction in length the **Adjacent** is to the **Hypotenuse**.

## Knowing one side, an Example to work out the second side of the Triangle

What we do now, is either get out our Trig tables, or plug into our calculator the following : **Cos 30 **= and we get the following answer : **.866 **( rounded down to** 3 decimal places** ). This tells us that the five inch length of the base of the **Triangle** is **.866** times that of the length of the** Hypotenuse**.

Now this means that if we knew this **Hypotenuse**, all we need do is multiply it by **.866** to find the length of the **Adjacent**, but seeing it is the other way round ( we know the **Adjacent**, and wish to find the **Hypotenuse** ), we do the opposite, and *divide* five inches by **.866**. This gives us **5.7735 " **, ( rounded down to four decimal places

**), which means that**

*if*the

**Adjacent**side to an angle of

**30°**is

**five**inches, then the

**Hypotenuse**will be

**5.7735**inches. You can also check this by multiplying

**5.7735**by

**.866**to get back to

**five**inches, to confirm that

**five**is

**.866**of

**5.7735**.

## Example to find the third side of the Triangle

This leaves us with the third side of the **Triangle**- the length of the edge opposite the angle of **30°**. Knowing the other two, we can now work out this third length a number of ways. First of all, let us see how this side relates to our **Adjacent** base of** five inches**. Now if we know the **Adjacent**, and want to find **the Opposite**, the obvious function to use is **the Tangent**, since as we know this is ratio of the length of the **Opposite** side divided by that of the **Adjacent**. What we can say here, is that the **Tangent** of** thirty degrees** equals the **length of the Opposite**, divided by **five**, or, that the length of the** Opposite** is equal to** five** times the **Tangent** of **30°**.** **

**Tan 30 **=** .57735**. Well, that is interesting, because we have seen something like that number before ! This is because all of this is in some way interconnected, which we shall see.

And, **5 **tan **30 **=** 2.887 **( **rounded up to 3 d.p. **)

This tells us that the relation between the** Opposite** and **Adjacent** of two sides meeting at **an angle of thirty degrees**, is that the **Opposite** is just *over half* ( **.5 **) of the length of the **Adjacent**. **Half of five inches is two and a half inches**, and **2.887** is just above that.

## How do sides relate ?

Now, even though we don’t need to, seeing we now have the lengths of all three sides, let us see how the** Opposite** relates to the **Hypotenuse**. The function that does this is **the Sine**, which is **the ratio we get when we divide the Opposite by the Hypotenuse**. The sine of **30° **equals **.5**, so what this tells us, is, that with respect to a**n angle of thirty degrees**, the length of the **Opposite** is exactly **.5**, or **half** that of the** Hypotenuse**. **2.887 **×** 2 **=** 5.774 **( remember that these numbers have been rounded up and down, so sometimes we get a bit of a rounding error )

## But some Ratios do have their Limitations

It is also needful to be aware, that both **Cosine** and** Sine** * never* reach a value above **one**, but **Tangent** can be any value. Also, if two angles add up to **ninety **degrees, the ** Sine** of one is the **Cosine** of the other, so that ** Sine** **30 ** =** Cosine 60**, and so forth. Also, it is useful to know that **the largest angle is always opposite the longest side**, and so** the longer the side opposite a given angle**, **the greater that angle will be**, and consequently, **the greater the angle**, **the larger the side opposite it can be**.

This stands to reason, if you imagine an angle acting like a kind of hinge - just as a door does - and this is the same sort of thing, which gives doors their usefulness. As you open a door, the gap between it and the door jamb closest to where the handle is when the door is shut, increases so that you can step through. But also, the angle formed at the door hinge end also increases.

Basically, the **Trigonometric Ratios**, i.e., **Tangent**, **Cosine** and **Sine** are **used to find angles if you know lengths**, and** lengths if you know angles**. If you know all three lengths, then you can find all the angles, but knowing all the angles won’t necessarily give you any of the lengths if you have no scale to work with, which is the principle behind **Similar Triangles**. To scale such a** Triangle**, therefore, You **need at least one length**, but once You have that length, with all three angles, You can then work out all of the rest.

## It's all Greek to Some

As to these lengths, we know that** for** **Right Angled Triangles**,** Pythgoras’ Theorem** **holds true**. We shall tackle him in the next** Hub**, as what he teaches us relates to **Trigonometry**,** ** The most basic of these **Triangles**, that is, the one with the shortest whole number values, where** all three sides are whole numbers**, is the **3 **:** 4 **:** 5 **triangle, where **3² **+** 4²** =** 5²**.

When I worked at a box factory here in **Christchurch** around 2006, one of the workers there told me that the way they used to construct a **Right Angle** was to build a **Triangle** with exactly these **three** measurements, (i.e., the **3 **:** 4 **:** 5 **triangle), since you knew that if you used these exact ones, **one of the angles would definitely be ninety degrees**, which they may need to build upright walls, or any other such thing.

Now, knowing all three of these sides, we can actually work out its interior angles. One of them, the largest, opposite the side that has a length of **five**, is **90°**. Let us draw the **Triangle** to see what we mean :

## Working out Angles of the 3 - 4 - 5

So let us now say we want the top angle, opposite the side that is** four inches long**. This will be the middle sized angle, and if we look, we see that if we take its** Opposite** ( =** 4 **), and divide this by the** Hypotenuse** ( =** 5 **), and from before we understand that** Opposite ÷ Hypotenuse** **= Sine**, and **4** ÷** 5 **=** .8**. So what we do is use our calculator to find the angle that has a ** Sine** equal to **.8**.

This we do by locating **the Arcus Sine Function** button. This is the button * falsely* symbolised by sin

^{-1 }on the calculator, which is carried out, depending on what type of machine you have, by some

**Inverse Function of the normal Sine button**. So,

**Arcus Sine**

**(**

**.8 ) = 53.13°**. That is to say, the

**Arcus Functions**of

**Sine**,

**Cosine**and

**Tangent**are

**always angles**- once again, the angles that You would have, whose

**Sine**,

**Cosine**or

**Tangent**, respectively, take the value given, for example here, just to be clear, that if the value is

**.8**, then

**.8**is the

**Sine**of some angle, and by using the

**Arcus Function**, we are trying to find the angle that

**.8**is the

**Sine**of. Thus from this answer, if the angle that

**.8**is the

**Sine**of is

**53.13°**, we can logically say that

**Sine**

**( 53.13°) = .8**.

Next, we could work the other angle out in a similar fashion, or we realise, that **this answer and our third angle add up to ninety**, so we subtract **53.13°** from **90°**, to get **36.87°**, or we do both to verify it - we add them up, then we also find the **Arcus Sine** value of **.6**, or** the Arcus Cosine of .8**, and all three values should correspond.

Now in terms of whole numbers, there are only a select few that work, so that the sum of the squares of the lesser two equal the square of the **Hypotenuse**. But if we were to multiply each of our lengths in the **3** :** 4 **:** 5 **triangle by the same number, we would end up with another** Right Angled Triangle**.

## To Conclude

For right angled triangles, there are two separate but very closely related topics, **Trigonometry**, which we have just had a good look at, and to come, **Pythagoras'** **Theorem**, where we in particular shall look at the concept touched upon here with the **3**:**4**:**5 **triangle, which is that of the **Pythagorean Triplet**.

## Disclaimer

As much as some of this Hub contains certain Mathematical knowledge accessible in the public domain, and not subject to any Copyright, other information has been drawn from textbooks which themselves are Copyright, but only in the sense of how they deliver the information which itself is shared and sometimes ancient Mathematical knowledge. Other information has also been found on Wikipedia ( Copyright 2013, the Wikimedia Foundation ), which is a good source of information. Part of this is also my own discovery, but may also independently have been found by others.

The term Trickonometry was, to my knowledge, invented by me, as a play on the words Trick and Trigonometry, in that this subject can indeed become quite tricky. With care and hard work, it can however be mastered. The quote about knowing one's Limitations can be seen to have been paraphrased from Dirty Harry's comments in the Movie Magnum Force ( 1973 ), but I am not sure if that is where I took it from. The **It's all Greek to some** quote is from Act One, Scene Two of Shakespeare's Julius Caesar, with Casca speaking to Cassius about what Cicero had said.

All illustrations in this particular Hub are my own, and have primarily been done using Microsoft™ Paintbox, by editing diagrams done originally on Microsoft™ Word. Any quote or part of this material which seems to belong to any other author should be treated as such, and I claim no ownership of anything I did not myself invent or discover, nor of any obvious copyright, trade mark, or registered trade mark.

The Adventure continues in the next Hub Pythagorean Theorem and Triplets,

If You are curious, then by all means do take a look at the other Hubs, The Maths They Never Taught Us - Part One, The Maths They Never Taught Us - Part Two , The Maths They Never Taught Us - Part Three, The Very Next Step - Squares and the Power of Two , And then there were Three - a Study on Cubes, Moving on to Higher Powers - a First look at Exponents, The Power of Many More - more on the Use of Exponents, Mathematics - the Science of Patterns , More on the Patterns of Maths , Mathematics of Cricket , The Shape of Things to Come , Things to do with Shapes, Pyramids - How to find their Height and Volume, How to find the Area of Regular Polygons, The Wonder and Amusement of Triangles - Part One, The Wonder and Amusement of Triangles - Part Two, the Law of Missing Lengths, The Wonder and Amusement of Triangles – Part Three : the Sine Rule, and The Wonder and Amusement of Triangles - Part Four : the Cosine Rule.

Also, feel free to check out my non Maths Hubs :

Bartholomew Webb , They Came and The Great New Zealand Flag

Just take a good look at it, and note how interesting it all is, then see if you can come up with anything else along the same lines. As usual, I would appreciate any comments, feedback and suggestions which would be given due credit, or indeed have a go and publishing Your ideas Yourselves, but firstly, by all means, add Your comments - it's a free Country.