definitions in trigonometry

Informal definitions

Given a triangle $ABC$ with a signed angle $x$ at $A$ and a right angle at $B$ , the ratios $$\frac{BC}{AC}\qquad \frac{AB}{AC}\qquad \frac{BC}{AB}$$ are dependent only on the angle $x$ , and therefore define functions, denoted by $$\sin x\qquad \cos x\qquad \tan x$$ respectively, where the names are short for sine, cosine and tangent. Their inverses are rather less important, but also have names: \begin{eqnarray*} \cot x &=& \frac{AB}{BC} = \frac{1}{\tan x} \text{ (cotangent)} \\ \csc x &=& \frac{AC}{BC} = \frac{1}{\sin x} \text{ (cosecant)} \\ \sec x &=& \frac{AC}{AB} = \frac{1}{\cos x} \text{ (secant)} \end{eqnarray*}From Pythagoras's theorem we have $\cos^2 x+\sin^2 x = 1$ for all (real) $x$ . Also it is ``clear'' from the diagram at left that functions $\cos$ and $\sin$ are periodic with period $2\pi$ . However:

Formal definitions

The above definitions are not fully rigorous, because we have not defined the word angle. We will sketch a more rigorous approach.

The power series $$\sum_{n=0}^\infty\frac{x^n}{n!}$$ converges uniformly on compact subsets of $\mathbb{C}$ and its sum, denoted by $\exp(x)$ or by $e^x$ , is therefore an entire function of $x$ , called the exponential function. $f(x)=\exp(x)$ is the unique solution of the boundary value problem $$f(0)=1\qquad f'(x)=f(x)$$ on $\mathbb{R}$ . The sine and cosine functions, for real arguments, are defined in terms of $\exp$ , simply by $$\exp(ix)=\cos x + i(\sin x)\;.$$ Thus $$\cos x = 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots$$ $$\sin x = \frac{x}{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$$ Although it is not self-evident, $\cos$ and $\sin$ are periodic functions on the real line, and have the same period. That period is denoted by $2\pi$ .