三角函数¶
定义¶
在\(Rt\Delta ABC\)中，设\(\angle ABC = \theta\)，同时我们定义一个角的邻边只包含直角边，则:
\[\sin \theta = \frac{\text{对边}}{\text{斜边}}, \cos \theta = \frac{\text{邻边}}{\text{斜边}}, \tan \theta = \frac{\text{对边}}{\text{邻边}}\]
\[\cot \theta = \frac{\text{邻边}}{\text{对边}}, \sec \theta = \frac{\text{斜边}}{\text{邻边}}, \csc \theta = \frac{\text{斜边}}{\text{对边}}\]
在此作者给出一些常见的三角函数取值：
| 角度 θ | \(\sin \theta\) | \(\cos \theta\) | \(\tan \theta\) | \(\cot \theta\) | \(\sec \theta\) | \(\csc \theta\) |
|---|---|---|---|---|---|---|
| \(0^\circ\) | 0 | 1 | 0 | \(\pm\infty\) | 1 | \(\pm\infty\) |
| \(30^\circ\) | \(\frac{1}{2}\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{\sqrt{3}}{3}\) | \(\sqrt{3}\) | \(\frac{2\sqrt{3}}{3}\) | 2 |
| \(\approx 37^\circ (3-4-5)\) | \(\frac{3}{5}\) | \(\frac{4}{5}\) | \(\frac{3}{4}\) | \(\frac{4}{3}\) | \(\frac{5}{4}\) | \(\frac{5}{3}\) |
| \(45^\circ\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{\sqrt{2}}{2}\) | 1 | 1 | \(\sqrt{2}\) | \(\sqrt{2}\) |
| \(\approx 53^\circ (3-4-5)\) | \(\frac{4}{5}\) | \(\frac{3}{5}\) | \(\frac{4}{3}\) | \(\frac{3}{4}\) | \(\frac{5}{3}\) | \(\frac{5}{4}\) |
| \(60^\circ\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{2}\) | \(\sqrt{3}\) | \(\frac{\sqrt{3}}{3}\) | 2 | \(\frac{2\sqrt{3}}{3}\) |
| \(90^\circ\) | 1 | 0 | \(\pm\infty\) | 0 | \(\pm\infty\) | 1 |