DT3 | Deriving Trigonometry - The Hexagon, and Pythagoras

So this third section has actually been covered by other posts on this blog - You may read up about the Magic Hexagon here, and you can read about the Pythagorean Identities here.

So this post is just to emphasise again how important these identites are. They can be derived as long as you have the knowledge and intuition of the lengths around a unit circle.

To recap, the Pythagorean identities are really important:

The Hexagon is a good way to remember these identities:

More interestingly, building on these concepts, it is possible to express every trigonometrical function in terms of every other.

Let's try to express sinθ in terms of all the other functions:

For cosine it's relatively easy:

For tangent, we can start with the basic identity we've learnt in the first section of Deriving Trigonometry:

This connects sine and tangent in one equation, but unfortunately also involves cosine, which we have to get rid of like this:

Then we can put this back into our equation with sine:

For cosecant it's very simple:

For secant, we can just use the result we got for cosine earlier and replace that in terms of secant:

As for cotangent, we can take the result we got from tangent and replace those with cotangent, and simplify from there:

As such, we have managed to express the sine function in terms of all other functions.

Here's the summary for sine:

And below is the table of all the six trigonometric functions expressed in terms of the other functions.

The next few sections of Deriving Trigonometry will build on these concepts and we will finally start to see identities that we are familiar with.

Stay tuned for more messy workings! :)