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Homedefinitions in trigonometry

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# 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:

$\displaystyle\cot x$ | $\displaystyle=$ | $\displaystyle\frac{AB}{BC}=\frac{1}{\tan x}\text{ (cotangent)}$ | ||

$\displaystyle\csc x$ | $\displaystyle=$ | $\displaystyle\frac{AC}{BC}=\frac{1}{\sin x}\text{ (cosecant)}$ | ||

$\displaystyle\sec x$ | $\displaystyle=$ | $\displaystyle\frac{AC}{AB}=\frac{1}{\cos x}\text{ (secant)}$ |

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

$f(0)=1\qquad f^{{\prime}}(x)=f(x)$ |

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 $2\pi$.

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## Attached Articles

complex sine and cosine by pahio

trigonometry by rm50

cosine at multiples of straight angle by pahio

sohcahtoa by Wkbj79

calculator trigonometric functions by Wkbj79

rigorous definition of trigonometric functions by CWoo

construction of tangent function from addition formula by rspuzio

derivatives of $\sin x$ and $\cos x$ by Wkbj79

trigonometric formulas from series by pahio