Fundamental Trigonometric Identities for the TMUA

Defining Trigonometric Identities

There are many ways to define trigonometric functions. Usually, we first meet trigonometry in relation to right-angled triangles. If we use this approach, the identities we use in the TMUA can be derived directly from the basic ratios of opposite, adjacent, and hypotenuse.

The Relationship between Tangent, Sine, and Cosine

It is sufficient for the TMUA to realise that the relationship between these three functions comes from the standard definitions provided when you first learn trigonometry:

1. $\sin \theta = \frac{O}{H}$

2. $\cos \theta = \frac{A}{H}$

3. $\tan \theta = \frac{O}{A}$

From these, it becomes obvious that $\tan \theta$ can be expressed as a ratio of $\sin \theta$ and $\cos \theta$:

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

This identity is particularly useful when you need to solve equations that contain both sine and cosine terms but no squared terms, as it allows you to convert the equation into one involving only tangent.

The Pythagorean Identity: $\sin^2 \theta + \cos^2 \theta = 1$

The second identity is fundamentally a version of Pythagoras' Theorem. Consider a right-angled triangle where the hypotenuse is 1 unit long. By the definitions of sine and cosine, the opposite side has length $\sin \theta$ and the adjacent side has length $\cos \theta$.

Applying Pythagoras' Theorem ($a^2 + b^2 = c^2$) to this triangle gives:

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

Which is written in standard notation as:

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

Universal Applicability and the CAST Diagram

While these identities are easily visualised using acute angles in a right-angled triangle, it is vital to notice that they apply to ANY angle $\theta$, whether it is acute, obtuse, reflex, or negative.

You can justify why these identities apply universally by considering the 'CAST' style diagrams. In these diagrams, the hypotenuse $H$ is always taken as a positive value (often $H = 1$). The signs of $\sin \theta$ and $\cos \theta$ change as the signs of the adjacent ($A$) and opposite ($O$) sides change across the four quadrants, but the relationships $\tan \theta = \frac{\sin \theta}{\cos \theta}$ and $\sin^2 \theta + \cos^2 \theta = 1$ remain consistent due to the algebraic properties of coordinates and the definition of a circle.

Practical Use in Solving Equations

In the TMUA, these identities are often used to reduce an equation to a single trigonometric function. For example, in equations involving $\cos^2 x$ and $\sin x$, you can substitute $\cos^2 x = 1 - \sin^2 x$. This creates a quadratic equation in terms of $\sin x$, which can then be factorised and solved using standard quadratic techniques.