1.5: 1.5 Trigonometric Functions

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Another extremely important group of functions are the fundamental trignonometric functions \(\sin\), \(\cos\), and \(\tan\). These can be defined in terms of the geometric ratios of the sides of right-angled triangles, as shown below:

If we use this basic definition, the domain is \(\theta \in [0, \,\pi/2)\), where the input angle \(\theta\) is given in radians.

We can generalize the definition using the following scheme, which allows for negative values of \(a\) and/or \(b\):

With this, the domain is extended to \(\theta \in [0,\,2\pi)\). We can further extend the domain to all real numbers, \(\theta \in \mathbb{R}\), by treating input values modulo \(2\pi\) as equivalent; in other words, \(f(\theta + 2\pi) = f(\theta)\). With this generalization, the trigonometric functions become many-to-one functions.

According to the Pythagorean theorem, \[\big[\sin(\theta)\big]^2 + \big[\cos(\theta)\big]^2 = 1.\] Using this, we can go on to prove a variety of identities, like the addition identities \[\begin{align} \sin(\theta_1 + \theta_2) &= \sin(\theta_1) \cos(\theta_2) + \cos(\theta_1)\sin(\theta_2) \\ \cos(\theta_1 + \theta_2) &= \cos(\theta_1) \cos(\theta_2) - \sin(\theta_1)\sin(\theta_2)\end{align}\] As you may recall, the trigonometric proofs for these identities involve drawing complicated triangle diagrams, cleverly applying the Pythagorean formula, etc. There are two problems with such proofs: (i) they require some ingenuity in the construction of the triangle diagrams, and (ii) it may not be obvious whether the proofs work if the angles lie outside \([0,\pi/2)\).

Happily, there is a solution to both problems. As we’ll soon see, such trigonometric identities can be proven algebraically by using complex numbers.