# 正弦定理

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

## 證明（一）

$\sin A = \frac{h}{b}$$\; \sin B = \frac{h}{a}$

$h = b\,\sin A = a\,\sin B$

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

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

## 证明（二）

### 角A为锐角时

$\ang \rm A = \ang D$

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

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

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

### 角A为直角时

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

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

### 角A为钝角时

$\angle \rm D = {\pi} - \ang A$

$\qquad\sin A = \sin D$

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

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

## 运用

### 三面角正弦定理

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

### 多边形的正弦关系

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

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

$\sin \ang OAB \sin \ang OBC \sin \ang OCD \sin \ang ODE \sin \ang OEA=\sin \ang OBA \sin \ang OCB \sin \ang ODC \sin \ang OED \sin \ang OAE$