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Trigonometry

Introductory Trigonometry

Author: oLahav

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Introduction to Trigonometry

Trigonometry, or "trig" as all the cool guys call it, is the branch of mathematics that deals with triangles, especially right-angled ones but regular ones too. It specifically looks for relationships between the sides and angles of triangles, using trig functions.

Let's start at the beginning:

This: is a triangle. It has 3 sides, and as a result 3 angles. We can call the angles by the letters next to them- A, B and C. We can call the sides by a lower-case letter representing the angle across from them- a, b and c.

In this particular triangle, say sides a and c are perpendicular- angle B is 90 degrees. This would make triangle ABC a right triangle. In right triangles, we call the long side B the hypotenuse. When we look at a particular angle, like A, the side across from that angle, A, is called opposite, and the other side, C, is called the adjacent.

An important thing about triangles is the sum of angles formula- all angles inside a triangle sum up to 180 degrees. Now, since ABC is right angled, we can see right away that A+C=90 degrees. So knowing either one of those angles will allow us to actually know all of the angles. Let's say angle A=53 degrees, so angle C would have to be 37 degrees.

So we have angles down, now what?

The cool thing about right triangles is that the ratio between the sides of 2 triangles with the same angles is always the same. So it doesn't really matter what sides you have, the ratios will always be equal.

Lucky for us, smart guys from ancient Egypt, India and Greece developed Trigonometric equations to identify these relations. Here they are:

$\cos A= \frac{adjacent}{hypotenuse}$ , $\sin A= \frac{opposite}{hypotenuse}$ , $\tan A= \frac{opposite}{adjacent}$ .

So, we've said A=53 degrees. This means that c/b= cos(53) = 0.6, a/b= sin(53) = 0.8, and a/c= tan(53)= 1.33. Now, if we knew one side, we could figure out all the rest!

Let's say side c is 3. Quick calculations would show that side a is 4, and side b is 5.

We can even use the Pythagorean Theorem for this: in our triangle, it would be a ^ 2 + b ^ 2= c ^ 2 .

Now, we'll look at trig functions

So, as we've seen, sin cos and tan are good for measuring ratios. But we can also use them in other ways, using them as functions.

Start with sin. You'll note that sin x can never be greater than 1, which is pretty clear through the Pythagorean Theorem- hypotenuse must be equal or greater than any of the other sides (same for cos!). We can extend the range of sin using the unit circle, with radians instead of degrees.

We get this kind of funky function:

This function can be extended beyond these coordinates, but it remains the same- it's periodic. The cos function works in a similar manner, only it's $\frac{\pi}{2}$ , or 90 degrees to the right.

Finally, the tan function isn't bounded by 1. However, it has no value at 90 degrees (clearly). It's also periodic, but it features discontinuous asymptotes. It looks like this:

And that's it for the functions. Nothing too big really.

Some basic identities:

You'll note quickly that $\tan \alpha=\frac{\sin \alpha}{\cos \alpha}$ . This is because, using a as the opposite, b as the hypotenuse and c as the adjacent, $\tan A=\frac{a}{c}=\frac{\frac{a}{b}}{\frac{c}{b}}=\frac{\sin A}{\cos A}$ . This is a useful identity.

My favourite identity is $\sin ^2 \alpha + \cos ^2 \alpha =1$ . This one makes sense because we already know that $\frac{a ^ 2+c ^2}{b ^ 2}=1$ , by the Pythagorean Theorem.

You can also mix and match these and other identities to prove lots of other trigonometric identities, such as $\tan \alpha + \sec \alpha = \tan (\frac{\alpha}{2} + \frac{\pi}{4})$ . But that's less important.