trigonometry axioms

Question

I remember as a college freshman having a math teacher present a short list of simple axioms from which we could then derive all trig functions and prove all identities. But I cannot find this list of axioms online. Perhaps I am using the wrong words. Can you help me?

Answer 1

It all begins with the Pythagorean Theorem: $a^2 + b^2 = c^2$, where $a$ and $b$ are the sides of a right triangle that aren't across from the $90^\circ$ angle and $c$ is the side that is across from that angle. This is a theorem, but, you can find the basis of trigonometry here. There are a lot of proofs of this theorem out there; I recommend to you to use Euclid's Method to prove it.

Then, let's figure out now:

-The sum of the angles of a right triangle is equal to $180^\circ$ or $\pi$ radians

-The right triangle has, then, 3 angles, a $90^\circ$ angle and another two such that the sum is $180^\circ$. Let's call them $\alpha$ and $\theta$ (Complementary angles).

-The sides of a right triangle that aren't across from a $90º$ angle are called "Legs".

-One is called the Adjacent Leg and the other the Opposite Leg, this depends of what angle we are talking about, $\alpha$ or $\theta$. The side with an angle of $90^\circ$ is called the Hypotenuse.

In this case you can see what is adjacent and what's opposite and why.

-There exist six trigonometric ratios:

$\sin(\theta)$, $\cos(\theta)$, $\tan(\theta)$, $\csc(\theta)$ , $\sec(\theta)$ , $\cot(\theta)$

And they are defined in this way

Let $\mathit{OL}$ be the Opposite Leg and $\mathit{AL}$ be the Adjacent Leg (with respect to $\theta$ in this case) and the Hypotenuse just $H$.

$\sin(\theta) = \frac{\mathit{OL}}H$ | $\cos(\theta) = \frac{\mathit{AL}}H$ | $\tan(\theta) = \frac{\mathit{OL}}{\mathit{AL}}$ | $\csc(\theta) = \frac{H}{\mathit{OL}}$ | $\sec(\theta) = \frac{H}{\mathit{AL}}$ | $\cot(\theta) = \frac{\mathit{AL}}{\mathit{OL}}$

You can see that, the Cosecant is the reciprocal of the Sine, the Secant of the Cosine and the Cotangent of the Tangent. This means that

$\sin(\theta)\csc(\theta)=1$

$\cos(\theta)\sec(\theta)=1$

$\tan(\theta)\cot(\theta)=1$

You can see too, that $\frac{\sin(\theta)}{\cos(\theta)} = \tan(\theta) $ and $\frac{\sec(\theta)}{\csc(\theta)} = \tan(\theta)$

The three fundamental identities of trigonometry:

$$\sin^2(\theta) + \cos^2(\theta) = 1$$

$$\tan^2(\theta) + 1 = \sec^2(\theta)$$

$$\cot^2(\theta) + 1 = \csc^2(\theta)$$

You can prove them by yourself just substituting the meaning for a triangle ($AL$, $OL$ and $H$).

For the last, I give to you, this representation of every trigonometric ratio:

Note: Imagine that the circle has a radius of $1$, and prove every identity.

Answer 2

You can find such a treatment from Michael Kassatly (Milton Academy) here.

The four axioms for trigonometry given there are as follows:

Axiomatic Trig:

1. $\sin(-x) = -\sin(x)$

2. $\sin(x+y) = \sin(x)\cos(y) + \cos(x)\sin(y)$

3. $\sin^2(x) + \cos^2(x) = 1$

4. $\sin(\pi/2) = 1$, $\cos(0) = 1$

The link continues with $44+$ identities that can be proved from these axioms.

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As a minor note, the fourth axiom can be slightly simplified: It is possible to prove that $\cos(0) = 1$ without that being given as information. One road to such a proof is as follows:

$\sin(0) = -\sin(0)$ by Axiom 1 means that $\sin(0) = 0$.

By Axioms 4 (the first half) and 2, we have:

$$1 = \sin(\pi/2) = \sin(\pi/2 + 0) = \sin(\pi/2)\cos(0) + \cos(\pi/2)\sin(0)$$

This final expression, combined additionally with $\sin(0) = 0$ as above, yields:

$$1\cdot\cos(0) + \cos(\pi/2)\cdot0 = \cos(0)$$

from which we conclude that $1 = \cos(0)$ as desired.