和角公式
圆是单位圆
- \(\displaystyle \sin \left( \alpha \pm \beta \right)=\sin \alpha \cdot \cos \beta \pm \sin \beta \cdot \cos \alpha\)
\(\because \angle HBD=\alpha\)
\(\therefore BD=\dfrac{1}{\cos \alpha} BH\)
\(\because \text{圆是单位圆}\)
\(\therefore BD=\dfrac{1}{\cos \alpha} 1 \times \sin \beta=\dfrac{\sin \beta}{\cos \alpha}\)
\(\begin{aligned} \therefore DE &= \sin \alpha \times OD \\ &=\sin \alpha \times (OH - DH) \\ &=\sin \alpha \times (BO \times \cos \beta - BD \times \sin \alpha) \\ &=\sin \alpha \times (\cos \beta - \dfrac{\sin \beta}{\cos \alpha} \times \sin \alpha) \end{aligned}\)
\(\begin{aligned}\therefore \sin(\alpha + \beta ) &= BD + DE \\ &= \sin \alpha \times (\cos \beta - \dfrac{\sin \beta}{\cos \alpha} \times \sin \alpha )+\dfrac{\sin \beta}{\cos \alpha} \\&= \sin \alpha \times \cos \beta - \sin^2 \alpha \times \dfrac{\sin \beta}{\cos \alpha}+\dfrac{\sin \beta}{\cos \alpha } \\ &= \sin \alpha \times \cos \beta + \dfrac{\sin \beta}{\cos \alpha} \times (1 - \sin^2 \alpha)\\ &= \sin \alpha \times \cos \beta + \dfrac{\sin \beta}{\cos \alpha} \times \cos^2 \alpha \\ &= \sin \alpha \times \cos \beta + \sin \beta \times \cos \alpha \end{aligned}\)
\(\because \sin \text{是奇函数},\cos \text{偶函数}\)
\(\therefore \sin \left(-x\right) = -\sin x , \cos (-x)= \cos x\)
\(\begin{aligned}\therefore \sin (\alpha - \beta) &= \sin (\alpha + (-\beta)) \\ &= \sin \alpha \times \cos (-\beta) + \sin (-\beta) \times \cos \alpha \\ &= \sin \alpha \times \cos \beta - \sin \beta \times \cos \alpha \end{aligned}\)
- \(\cos (\alpha \pm \beta) = \cos \alpha \times \cos \beta \mp \sin \alpha \times\sin \beta\)
同上列出
\(\therefore BD=\dfrac{\sin \beta}{\cos \alpha}\)
\(\therefore DE =\sin \alpha \times (\cos \beta - \dfrac{\sin \beta}{\cos \alpha} \times \sin \alpha)\)
\(\begin{aligned} \therefore \cos (\alpha + \beta) &= \dfrac{OE}{BO}\\ &= DE \times \dfrac{\cos \alpha}{\sin \alpha} \\ &= \sin \alpha \times (\cos \beta - \dfrac{\sin \beta}{\cos \alpha} \times \sin \alpha) \times \dfrac{\cos \alpha}{\sin \alpha} \\ &= \cos \alpha \times \cos \beta - \sin \alpha \times \sin \beta \end{aligned}\)
\(\because \sin \text{是奇函数},\cos \text{偶函数}\)
\(\therefore \sin \left(-x\right) = -\sin x , \cos (-x)= \cos x\)
\(\begin{aligned} \cos (\alpha - \beta) &= \cos (\alpha + (-\beta)) \\ &= \cos \alpha \times \cos (-\beta) - \sin \alpha \times \sin (-\beta) \\ &= \cos \alpha \times \cos \beta + \sin \alpha \times \sin \beta \end{aligned}\)