Trigonometry: Simplified Part II

If you haven’t read Part I recently, I highly suggest you give it a quick read through to re-familiarise yourself with our goals and intentions for this part directly following from Part I.

Previously, we discovered the following results about scaling triangles:

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

and was then motivated to take the crucial step into answering the following question:

Do triangles that are scaled differently have the same angle AND the same ratio of sides?

Before doing so however, it would be best to define a bit some terms for us so that we may more easily mention them later on. What we shall do is name the sides of the triangle based on where the angle is.

—-Definitions—-

What we do is call the opposite side to the angle Q as ‘Opposite’, the side directly opposite the right-angle as ‘Hypotenuse’ and the remaining side (or the side next to the angle Q) as ‘Adjacent’.

QUICK FIRE: HOW TO QUICKLY LABEL ANY TRIANGLE:

1. Pick an angle in the triangle
2. Call the opposite side to it ‘Opposite’
3. Call the side opposite the right angle as the ‘Hypotenuse’
4. Call the remaining side ‘Adjacent’

Recall that:

“Triangles that are scaled differently have the same ratio of sides”

So now in hopes to name every type of ratio we can, we question how many ratios are we able to make?

As can be seen, we can make a total of 3 DISTINCT ratios, and no other combination of sides will give something new.

$1.$  $\frac {opposite}{hypotenuse}$

$2.$  $\frac {adjacent}{hypotenuse}$

$3.$  $\frac {opposite}{adjacent}$

To not only depend on numbers 1, 2 and 3, we shall be giving these ratios names:

$3.$ Tangent ratio $= \frac {opposite}{adjacent}$

Now I know that at this stage with the wave of definitions and terms, you might begin to lose track of what is what. To aide this, here is a bit of a reminder sheet of what is what.

—-End of Definitions—-

Now, remember that we wanted to answer the following question:

Do triangles that are scaled differently have the same angle AND the same ratio of sides?

This essentially boils down to the question

Is every angle uniquely associated with a ratio of sides?

After all, suppose it was the case that 2 DIFFERENT angles produced the SAME ratio. This would mean that it would NOT be true that EVERY triangle have the SAME angle and the SAME ratio, since we have found a triangle which would have a ratio associated with 2 angles!

To show this, let us fix the Hypotenuse of our triangle to be equal to length 1, and let us vary our angle Q. By doing this:

$2.$ Cosine ratio $= \frac {adjacent}{1} = adjacent$

But what does this mean exactly? If we look at our reference triangle, the opposite side is just the height of the triangle and the adjacent side is just the base of the triangle.

Therefore it would mean that our Sine ratio is just the length of the height of the triangle, and the Cosine ratio is just the length of the base of the triangle.

All that we do now is check whether as we vary the angle Q of the triangle, that we never come across triangles which have the same height or base. But if we vary the angle Q and keep the hypotenuse fixed, what does this trace out? A circle of fixed radius 1! (or at least a quarter of it)

In this case, ‘A’ stands for ANGLE, ‘B’ stands for BASE and ‘H’ stands for HEIGHT.

You can notice that at no point do we ever obtain 2 different heights/bases from seperate triangles that have the same angle, nor do we ever obtain 2 different angles from seperate triangles of the same height/base.

We’ve done it! We have shown the rather amazing fact that

Every triangle, regardless of scaling, is associated with a unique angle and a unique ratio of sides

This means, that the ratio of the sides of a triangle, is very DEPENDENT on the angle of a triangle, and vice versa. Due to this fact, we alter our above definitions to include this:

Note: Even though I didn’t explicitly state it, all the above holds as well for the Tangent Ratio

Now to avoid the hassle of writing too much (because we are lazy beings after all)
We are going to simplify our reminder sheet by replacing ‘Sine Ratio’ with ‘sin’, ‘Cosine Ratio’ with ‘cos’ and ‘Tangent Ratio’ with ‘tan’ to achieve:

Which are the infamous equations for trigonometry!

If you have understood everything up till now, congratulations! You have simplified your idea of the insurmountable and inescapable subject from hell you previously knew as Trigonometry, and replaced it with a playful manipulation of triangles and angles.