# trigonometric functions

## definitions

六个三角函数. 设直角三角形中, $\theta$ 的对边、邻边和斜边分别是
$a$, $b$, $c$, 得到
-   sine. $$\sin\theta = \frac{a}{c}$$
-   cosine. $$\cos\theta = \frac{b}{c}$$
-   tangent. $$\tan\theta = \frac{a}{b}$$
-   cotangent. $$\cot\theta = \frac{b}{a}$$
-   secant. $$\sec\theta = \frac{c}{b}$$
-   cosecant. $$\csc\theta = \frac{c}{a}$$

三个不同适用范围的定义.
-   直角三角形定义. 根据此定义, $\theta \in [0, \frac{\pi}{2}]$.
-   单位圆定义. 圆心在原点的单位圆上点的座标为 $(\cos \theta, \sin \theta)$.
    根据此定义, $\theta \in \mathbb{R}$. 该定义与双曲函数的双曲线定义对应.
-   Taylor 级数定义. 根据此定义, $\theta \in \mathbb{C}$.
\begin{align}
\sin x &= \sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!}\\
\cos x &= \sum_{n=0}^\infty \frac{(-1)^nx^{2n}}{(2n)!}
\end{align}

## geometric relations
两种不同的单位圆表示方法.


曲线和值的关系.


## properties

-   关系.
    \begin{align}
    \tan\theta &= \frac{\sin\theta}{\frac\cos\theta}\\
    \cot\theta &= \frac{1}{\tan\theta}\\
    \sec\theta &= \frac{1}{\cos\theta}\\
    \csc\theta &= \frac{1}{\sin\theta}
    \end{align}
-   周期性. (三角函数的单位圆定义的必然推论.)
    $\tan$ and $\cot$ 的周期是 $k\pi$, 其他 4 个周期是 $2k\pi$.
-   复数域上的性质. 由 $\sin x$, $\cos x$, $e^x$ 的 Taylor expansion, 可以得到 Euler's
    formula 成立.
    $$e^{ix} = \cos x + i \sin x$$
    $$\cos x = \operatorname{Re}(e^{ix}), \sin x = \operatorname{Im}(e^{ix})$$
    $$\sin x = \frac{e^{ix} - e^{-ix}}{2i}, \cos x = \frac{e^{ix} + e^{-ix}}{2}$$
    complex sine, cosine 和 real sine, cosine 以及 real hyperbolic sine, cosine
    的关系
    \begin{align}
    \sin(x+iy) &= \sin x \cosh y + i \cos x \sinh y\\
    \cos(x+iy) &= \cos x \cosh y - i \sin x \sinh y
    \end{align}

## etymology
-   sine. Latin sinus for gulf or bay, means "open".
-   cosine. The sine of the complementary or co-angle.
-   tangent. So called because it can be represented as a line segment tangent
    to the circle, that is the line that touches the circle but not cuts it.
    从图中看得很清楚.
-   cotangent. Tangent of co-angle.
-   secant. So called because it represents the line that cuts the circle (from
    Latin: secare, to cut). 从图中看得很清楚.
-   cosecant. The secant of co-angle.




```python

```

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2021-05-12T08:46:18.000Z
math/special-functions.ipynb
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2019-02-13T03:11:33.000Z
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Naitreey/notes-and-knowledge
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BSD-3-Clause
2019-04-11T11:02:32.000Z
2020-06-27T11:59:09.000Z
Qwen/Qwen-72B
1. YES
2. YES