| Version 4 |
Version 3 |
| \PMlinkescapeword{terms} |
\PMlinkescapeword{terms} |
| \PMlinkescapeword{names} |
\PMlinkescapeword{names} |
| \PMlinkescapeword{inverses} |
\PMlinkescapeword{inverses} |
| \includegraphics{trig.eps} |
\includegraphics{trig.eps} |
|
|
| \textbf{Informal definitions} |
\textbf{Informal definitions} |
|
|
| Given a triangle $ABC$ with a signed angle $x$ at $A$ and a |
Given a triangle $ABC$ with a signed angle $x$ at $A$ and a |
| right angle at $B$, the ratios |
right angle at $B$, the ratios |
| $$\frac{BC}{AC}\qquad \frac{AB}{AC}\qquad \frac{BC}{AB}$$ |
$$\frac{BC}{AC}\qquad \frac{AB}{AC}\qquad \frac{BC}{AB}$$ |
|
are dependent only on the angle $x$, and therefore define functions,
|
are dependant only on the angle $x$, and therefore define functions,
|
| denoted by |
denoted by |
| $$\sin x\qquad \cos x\qquad \tan x$$ |
$$\sin x\qquad \cos x\qquad \tan x$$ |
| respectively, where the names are short for \emph{sine, cosine} and |
respectively, where the names are short for \emph{sine, cosine} and |
| \emph{tangent}. Their inverses are rather less important, |
\emph{tangent}. Their inverses are rather less important, |
| but also have names: |
but also have names: |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| \cot x &=& AB/BC = \frac{1}{\tan x} \text{ (cotangent)} \\ |
\cot x &=& AB/BC = \frac{1}{\tan x} \text{ (cotangent)} \\ |
| \csc x &=& AC/BC = \frac{1}{\sin x} \text{ (cosecant)} \\ |
\csc x &=& AC/BC = \frac{1}{\sin x} \text{ (cosecant)} \\ |
| \sec x &=& AC/AB = \frac{1}{\cos x} \text{ (secant)} |
\sec x &=& AC/AB = \frac{1}{\cos x} \text{ (secant)} |
| \end{eqnarray*} |
\end{eqnarray*} |
| From Pythagoras's theorem we have $\cos^2 x+\sin^2 x = 1$ for all (real) $x$. |
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 $\sin$ |
Also it is ``clear'' from the diagram at left that functions $\cos$ and $\sin$ |
| are periodic with period $2\pi$. However: |
are periodic with period $2\pi$. However: |
|
|
| \textbf{Formal definitions} |
\textbf{Formal definitions} |
|
|
| The above definitions are not fully rigorous, because we have not defined |
The above definitions are not fully rigorous, because we have not defined |
| the word \emph{angle}. We will sketch a more rigorous approach. |
the word \emph{angle}. We will sketch a more rigorous approach. |
|
|
| The power series |
The power series |
| $$\sum_{n=0}^\infty\frac{x^n}{n!}$$ |
$$\sum_{n=0}^\infty\frac{x^n}{n!}$$ |
| converges uniformly on compact subsets of $\mathbb{C}$ and its sum, |
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 $x$, |
denoted by $\exp(x)$ or by $e^x$, is therefore an entire function of $x$, |
| called the exponential function. |
called the exponential function. |
| $f(x)=\exp(x)$ is the unique solution of the boundary value problem |
$f(x)=\exp(x)$ is the unique solution of the boundary value problem |
| $$f(0)=1\qquad f'(x)=f(x)$$ |
$$f(0)=1\qquad f'(x)=f(x)$$ |
| on $\mathbb{R}$. |
on $\mathbb{R}$. |
| The sine and cosine functions, for real arguments, are defined in terms |
The sine and cosine functions, for real arguments, are defined in terms |
| of $\exp$, simply by |
of $\exp$, simply by |
| $$\exp(ix)=\cos x + i(\sin x)\;.$$ |
$$\exp(ix)=\cos x + i(\sin x)\;.$$ |
| Thus |
Thus |
| $$\cos x = 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\ldots$$ |
$$\cos x = 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\ldots$$ |
| $$\sin x = \frac{x}{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}-\ldots$$ |
$$\sin x = \frac{x}{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}-\ldots$$ |
| Although it is not self-evident, $\cos$ and $\sin$ are periodic functions on |
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 $2\pi$. |
the real line, and have the same period. That period is denoted by $2\pi$. |
