三角函数诱导公式

推导原理

①三角形内角和180°
②y值是线段OA投影到周的移动距离,即AC⊥x
③平面几何中的坐标正负

1.$2k\Pi$

线移动2k+θ后

• 线与x的夹角未发生变化
• 投影x轴位置未变化
• 投影y轴位置未变化
  $$sin(2k+\theta )=sin(\theta ),k\in Z$$
  $$cos(2k+\theta )=cos(\theta ),k\in Z$$
  $$tan(2k+\theta )=tan(\theta ),k\in Z$$
  $$cot(2k+\theta )=cot(\theta ),k\in Z$$

2.$2k\Pi +\frac{\Pi }{2}$

线移动$2k\Pi +\frac{\Pi }{2}+\theta$后

• 线与x轴夹角变换,即$\frac{\Pi }{2}-\theta$
• 投影到x的+轴
• 投影y的-轴
  $$sin(\frac{\Pi }{2}+\theta )=cos(\theta ),k\in Z$$
  $$cos(\frac{\Pi }{2}+\theta )=sin(\theta ),k\in Z$$
  $$tan(\frac{\Pi }{2}+\theta )=cot(\theta ),k\in Z$$
  $$tan(\frac{\Pi }{2}+\theta )=tan(\theta ),k\in Z$$

3.$2k\Pi +\Pi$

线移动$2k\Pi +\Pi$后

• 线与x轴发生夹角未发生变换
• 投影x的-轴
• 投影y的-轴
  $$sin(2k\Pi +\Pi +\theta )=sin(\theta ),k\in Z$$
  $$cos(2k\Pi +\Pi +\theta )=-cos(\theta ),k\in Z$$
  $$tan(2k\Pi +\Pi +\theta )=-tan(\theta ),k\in Z$$
  $$cot(2k\Pi +\Pi +\theta )=-cot(\theta ),k\in Z$$

4.$2k\Pi +\frac{3\Pi }{2}$

线移动$2k\Pi +\frac{3\Pi }{2}$后

• 线与x轴发生夹角未生变换,即$\frac{\Pi }{2}-\theta$
• 投影x的-轴
• 投影y的+轴
  $$sin(2k\Pi +\frac{3\Pi }{2}+\theta )=-cos(\theta ),k\in Z$$
  $$cos(2k\Pi +\frac{3\Pi }{2}+\theta )=sin(\theta ),k\in Z$$
  $$tan(2k\Pi +\frac{3\Pi }{2}+\theta )=-cot(\theta ),k\in Z$$
  $$cot(2k\Pi +\frac{3\Pi }{2}+\theta )=-tan(\theta ),k\in Z$$

5.-θ

-θ表示线移动角度由正方向变换正方向移动
$$sin(-\theta )=-sin(\theta )$$
$$cos(-\theta )=cos(\theta )$$
$$tan(-\theta )=-tan(\theta )$$
$$cot(-\theta )=-cot(\theta )$$