In this post I’ll be proving to you that:

Now firstly I will have to say that:

And also that:

If this is the case, then…

Since this is in the form:

I would have to conclude that:

Hence I’ve proven that:

In this post I’ll be proving why:

Let’s say that:

And also that:

This would imply that:

Now if we multiply and together, we get:

Which is thanks to what we know about **trigonometric identities**.

As we can see above, we’ve formed another complex number:

And this is in the form of:

And because of the rules of complex numbers, we can say that:

Hence, we have our proof.

Hello. In this post I’ll be showing you how to derive sin(0°), sin(15°), sin(30°), sin(45°), sin(60°), sin(75°), sin(90°), cos(0°), cos(15°), cos(30°), cos(45°), cos(60°), cos(75°), cos(90°), tan(0°), tan(15°), tan(30°), tan(45°), tan(60°), tan(75°) and tan(90°) from absolute scratch.

Now, I’ll first start off by showing you how to derive sin(30°), sin(60°), cos(30°) and cos(60°) with the use of an **equilateral triangle** (image above). This equilateral triangle has lengths equal to 2. If you look at the diagram above and its properties carefully, you should conclude that:

Alright, so far so good. Next, have a look at this **isosceles triangle** (image above). If you take its properties into consideration – you’ll discover that:

Ok, so I’ve already shown you how to derive sin(30°), sin(45°), sin(60°), cos(30°), cos(45°) and cos(60°) using simple diagrams. It turns out that **with the information above** and also some **trigonometric identities** – we can derive sin(15°), sin(75°), cos(15°) and cos(75°). Let me show you what I mean…

sin(0°), sin(90°), cos(0°) and cos(90°) are values you should already know, so I won’t be demonstrating how to derive them. If you have studied the **unit circle** – you’ll know that:

These values are fairly easy to find.

So, this is the moment you’ve been waiting for… The complete set of derivations I said I’d give you. Although it may seem hard to derive tan(0°), tan(15°), tan(30°), tan(45°), tan(60°), tan(75°) and tan(90°) from absolute scratch, or like a tedious task – we have already done most of the hard work. All these tangent values can be derived using the information we’ve already accumulated, because:

Therefore:

And now, the set of derivations is complete. 😀