Trigonometry: Simplified Part I

Trigonometry. It is the bane of most people’s existence when it comes to their early mathematical education. People look down upon this subject for being tedious, unintuitive, boring and most importantly of all, useless. These individuals have very good reasons to believe so. After all, the way mathematics is taught in some areas of the world does not differ much from what they have described. But then what’s unintuitive and tedious is not the subject that is taught, but rather the method in which it was taught! In hopes to repair the misunderstood view that trigonometry is boring, and that therefore all of mathematics must be, I shall be giving you a rundown for the motivation behind trigonometry, as well as an intuitive understanding for why and how it came to be. Skipping of course the 100s of different exercises to find the missing side.

Let’s start off with a right-angled triangle, which is a triangle with one of its angles 90^{\circ} and has sides which we decide to randomly name a, b, c . We shall call this triangle ABC:

right

Let’s now think of another triangle like the one below where we now label the sides x, y, z and call the triangle XYZ.

right2

These appear to look like the EXACT same right triangle as before. But in fact, I have actually played a little trick on you with the scaling of the images because if we put these triangles next to each other:

right3

What’s the point of this you might ask? Well, the point is to recognize that these triangles are in reality, the exact same triangle, just scaled slightly differently from one another. To prove this to you, we shall demonstrate how the angles of one of these triangles matches up exactly with the other. Notice that one of the angles is already the same, that is the right angle. For the remaining two:

ang2.png

As you can see, both triangles have the same angles, which makes them the same exact triangle just ‘scaled’ differently. Yet, even though the length of the individual triangle’s sides are different, there is an aspect which doesn’t change, no matter how you change the scale. What appears to remain the same regardless of scaling is the ratio of the sides of the triangle. That is the one of the main purposes of uses ratios in Trigonometry.

We can very easily show that the ratio remains the same under scaling by taking the ratio of the sides x:y . If we change the scale of the triangle, say by half, then we now half the length of the sides to be \frac {x}{2} , \frac {y}{2} . Thus, their ratio would be \frac {x}{2} : \frac {y}{2} but this is the exact same ratio as x:y from multiplying both sides by 2!

ratio2.png

From this we have shown that the ratio of the sides remains constant, despite you scaling the triangle up or down.

So what have we concluded up till now in this short space of time?

  • Triangles that are scaled differently have the same angle
  • Triangles that are scaled differently have the same ratio of sides

Wouldn’t it be great if we can take one more step and combine these two statements to say that:

  • Triangles that are scaled differently have the same angle AND the same ratio of sides?

This would mean that every ANGLE of a triangle is uniquely linked with a certain RATIO OF SIDES of the triangle and more importantly,

THE SCALING OF THE TRIANGLE DOES NOT MATTER

This would make solving problems in Trigonometry a somewhat trivial task, because if you know one, you automatically know the other!

The only problem that we face is how to properly show this property. Before we do so however, we must first encounter some definitions in Part II and from there see how this naturally comes about in a beautiful manner!

To wrap back to our original question, a standard question in Trigonometry would ask for the missing side in the figure below:

unk

This question is essentially asking “What is the ratio of the sides of the triangle?”, but since the length of one of the sides is missing, we do not know what this! But from what we have already discussed, if we know the angle, we should automatically know the ratio! From there, finding the length of the side becomes a question of bringing the unknown subject of the formula.

This is the essence of TRIGONOMETRY.

Thanks for reading!

-Sinthorel

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