Fundamental Trigonometric Identities

Trigonometric identities

Knowledge and use of

$\tan \theta = \frac{\sin \theta}{\cos \theta}$

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

There are many ways to define trigonometric functions. Usually, we first meet trigonometry in relation to right-angled triangles, and if we use this approach then the formula $\sin^2 \theta + \cos^2 \theta = 1$ is clearly just a version of Pythagoras' Theorem. We have drawn a diagram to illustrate this.

You should notice that whilst we have drawn a triangle with an acute angle $\theta$, the formula applies to ANY angle. Can you justify why it must apply to ANY angle using the "CAST" style diagrams we drew earlier?

In addition, you should know that $\tan \theta = \frac{\sin \theta}{\cos \theta}$ and it is sufficient for the TMUA/ESAT to realise this comes from the standard definition of trigonometric functions you tend to be told when you first learn them:

$\sin \theta = \frac{O}{H} \quad \cos \theta = \frac{A}{H} \quad \tan \theta = \frac{O}{A}$

So, it is "obvious" that

$\tan \theta = \frac{O}{A} = \frac{O/H}{A/H} = \frac{\sin \theta}{\cos \theta}$

You should make sure that you understand how the signs of the three trigonometric functions change as the signs of $A$ and $O$ change in CAST diagrams [and note that CAST diagrams always have the $H$ value as positive, usually we set $H = 1$].