# 正弦定理

${\displaystyle {\frac {a}{\sin \angle A}}={\frac {b}{\sin \angle B}}={\frac {c}{\sin \angle C}}=2R}$

## 證明一

${\displaystyle \sin A={\frac {h}{b}}}$

${\displaystyle \;\sin B={\frac {h}{a}}}$

${\displaystyle h=b\,\sin A=a\,\sin B}$

${\displaystyle {\frac {\sin A}{a}}={\frac {\sin B}{b}}}$

${\displaystyle {\frac {\sin B}{b}}={\frac {\sin C}{c}}}$

## 證明二

${\displaystyle \triangle ABC}$的外接圓，設半徑為${\displaystyle R}$${\displaystyle BC=a}$

### 角A為銳角時

${\displaystyle \angle {\rm {A=\angle D}}}$

${\displaystyle {\rm {BD}}=2R,\ \angle {\rm {BCD}}={\pi \over 2}}$

${\displaystyle \sin \angle D={a \over 2R}}$
${\displaystyle \sin \angle A={a \over 2R}}$
${\displaystyle {a \over \sin \angle A}=2R}$

### 角A為直角時

${\displaystyle \sin \angle A=\sin {\pi \over 2}=1}$

${\displaystyle {a \over \sin \angle A}=2R}$

### 角A為鈍角時

${\displaystyle \angle {\rm {D={\pi }-\angle BAC}}}$

${\displaystyle \qquad \sin \angle BAC=\sin \angle D}$

${\displaystyle {\sin \angle BAC}={\sin \angle D}={a \over 2R}}$

${\displaystyle {a \over \sin \angle BAC}=2R}$

${\displaystyle {\frac {a}{\sin \angle A}}={\frac {b}{\sin \angle B}}={\frac {c}{\sin \angle C}}=2R}$

## 運用

### 三面角正弦定理

${\displaystyle {\frac {\sin \alpha }{\sin A}}={\frac {\sin \beta }{\sin B}}={\frac {\sin \gamma }{\sin C}}}$[1]

### 多邊形的正弦關係

${\displaystyle {\frac {OA}{\sin \angle OBA}}={\frac {OB}{\sin \angle OAB}},{\frac {OB}{\sin \angle OCB}}={\frac {OC}{\sin \angle OBC}},{\frac {OC}{\sin \angle ODC}}={\frac {OD}{\sin \angle OCD}},{\frac {OD}{\sin \angle OED}}={\frac {OE}{\sin \angle ODE}},{\frac {OE}{\sin \angle OAE}}={\frac {OA}{\sin \angle OEA}}}$

${\displaystyle {\frac {\sin \angle OAB\sin \angle OBC\sin \angle OCD\sin \angle ODE\sin \angle OEA}{\sin \angle OBA\sin \angle OCB\sin \angle ODC\sin \angle OED\sin \angle OAE}}={\frac {OB\cdot OC\cdot OD\cdot OE\cdot OA}{OA\cdot OB\cdot OC\cdot OD\cdot OE}}=1}$

${\displaystyle \sin \angle OAB\sin \angle OBC\sin \angle OCD\sin \angle ODE\sin \angle OEA=\sin \angle OBA\sin \angle OCB\sin \angle ODC\sin \angle OED\sin \angle OAE}$