三角関数の公式の備忘録

概要

Positional Encodingの解釈を説明していたブログにいい内容があったのでメモ
積和の公式などは完全に忘れているし、三角関数は大人になっても以外に使うので残す
例えば、ニューラルネットワークでは活性化関数としてtanhは良く利用されていたりする

基本

正弦、余弦、正接

三角形には辺が3つあるため、そこから2辺を選んでx/yのようなかたちで書く選び方は3×2 = 6通り
sin, cos, tanで3通り出ているため、残りは3通りで、それらがcose, sec, cot

$$ \cosec \theta = \frac{c}{a} = \frac{1}{sin \theta} \\\\ \sec \theta = \frac{c}{b} = \frac{1}{cos \theta} \\\\\ \tan \theta = \frac{{\sin \theta}}{{\cos \theta}} \\\\ \cot \theta = \frac{b}{a} = \frac{1}{tan \theta} = {\tan ^{ – 1}}\theta = \frac{{\cos \theta}}{{\sin \theta}} $$

ラジアンと度数の変換（Radian and degree conversion）

$$ \frac{b}{\pi } = \frac{\alpha }{{180}} $$

符号（Sign）

1
2
3
4
5
Quadrant 1 2 3 4
sin      + + - -
cos      + - - + 
tan      + - + -
cot      + - + -

オイラーの公式

オイラーの公式で次のように表現する。

$$ e^{ix} = \cos x + i \sin x \\\\ sin x = \frac{e^{ix}-e^{-ix}}{2i} \\\\ cos x = \frac{e^{ix}+e^{-ix}}{2} $$

双曲線関数（hyperbolic function）

指数関数$e^x$をもとに定義される以下は双曲線関数。

$$ \sinh x = \frac{e^{x}-e^{-x}}{2} \\\\ \cosh x = \frac{e^{x}+e^{-x}}{2} \\\\ \tanh x = \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} $$

双曲線関数のグラフ

$y=\cosh x$ のグラフは紫
$y=\sinh x$ のグラフは赤
$y=\tanh x$ のグラフは緑

二乗の公式（square formula）

$$ \sin^2 x = \frac{1 - \cos(2x)}{2} \\\\ \cos^2 x = \frac{1 + \cos(2x)}{2} \\\\ \sin^2 x + \cos^2 x = 1 $$

加法定理（Addition theorems）

$$ \begin{array}{l} \sin \left( {\alpha + \beta } \right) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \\\\ \sin \left( {\alpha – \beta } \right) = \sin \alpha \cos \beta – \cos \alpha \sin \beta \\\\ \cos \left( {\alpha + \beta } \right) = \cos \alpha \cos \beta – \sin \alpha \sin \beta \\\\ \cos \left( {\alpha – \beta } \right) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \end{array} $$$$ \begin{array}{l} \tan \left( {\alpha + \beta } \right) = \frac{{\tan \alpha + \tan \beta }}{{1 – \tan \alpha \tan \beta }} \\\\ \tan \left( {\alpha – \beta } \right) = \frac{{\tan \alpha – \tan \beta }}{{1 + \tan \alpha \tan \beta }} \\\\ \cot \left( {\alpha + \beta } \right) = \frac{{\cot \alpha \cot \beta – 1}}{{\cot \beta + \cot \alpha }} \\\\ \cot \left( {\alpha – \beta } \right) = \frac{{\cot \alpha \cot \beta + 1}}{{\cot \beta – \cot \alpha }} \end{array} $$

相互関係（Relations）

$$ \begin{array}{l} {\sin ^2}x + {\cos ^2}x = 1 \\\\ {\tan ^2}x + 1 = \frac{1}{{{{\cos }^2}x}}\\\\ {\cot ^2}x + 1 = \frac{1}{{{{\sin }^2}x}}\\\\ \cot x = \frac{1}{{\tan x}} \end{array} $$

和積の公式（Sum to product）

$$ \begin{array}{l} \sin \alpha + \sin \beta = 2\sin \frac{{\alpha + \beta }}{2}\cos \frac{{\alpha – \beta }}{2}\\\\ \sin \alpha – \sin \beta = 2\sin \frac{{\alpha – \beta }}{2}\cos \frac{{\alpha + \beta }}{2}\\\\ \cos \alpha + \cos \beta = 2\cos \frac{{\alpha + \beta }}{2}\cos \frac{{\alpha – \beta }}{2}\\\\ \cos \alpha – \cos \beta = – 2\sin \frac{{\alpha + \beta }}{2}\sin \frac{{\alpha – \beta }}{2} \end{array} $$

積和の公式（Product to sum)

$$ \begin{array}{l} \sin \alpha \sin \beta = – \frac{1}{2}\left( {\cos \left( {\alpha + \beta } \right) – \cos \left( {\alpha – \beta } \right)} \right) \\\\ \sin \alpha \cos \beta = \frac{1}{2}\left( {\sin \left( {\alpha + \beta } \right) + \sin \left( {\alpha – \beta } \right)} \right) \\\\ \cos \alpha \cos \beta = \frac{1}{2}\left( {\cos \left( {\alpha + \beta } \right) + \cos \left( {\alpha – \beta } \right)} \right) \end{array} $$

アーク関数（Arcus functions）

-1乗かarcxxxで表現する。

$$ \arcsin x = \sin^{-1}x \\\\ \arccos x = \cos^{-1}x \\\\ \arctan x = \tan^{-1}x \\\\ $$$$ \begin{array}{l} \sin \left( {\arcsin a} \right) = \cos \left( {\arccos a} \right) = a,a \in \left[ { – 1,1} \right]\\\\ \tan \left( {\arctan a} \right) = \cot \left( {{\mathop{\rm arccot}\nolimits} a} \right) = a,a \in\left[ { – \infty,\infty} \right] \end{array} $$

Arcus values

$$ \begin{array}{l} \alpha = \arcsin a \Leftrightarrow a = \sin \alpha ,\alpha \in \left[ { – \frac{\pi }{2},\frac{\pi }{2}} \right]\\\\ \beta = \arctan a \Leftrightarrow a = \tan \beta ,\beta \in \left[ {\frac{{ – \pi }}{2},\frac{\pi }{2}} \right]\\\\ \gamma = \arccos a \Leftrightarrow a = \cos \gamma ,\gamma \in \left[ {0,\pi } \right]\\\\ \varphi = {\mathop{\rm arccot}\nolimits} a \Leftrightarrow a = \cot \varphi ,\varphi \in \left[ {0,\pi } \right] \end{array} $$

Arcsin of sin

$$ \arcsin \left( {\sin \alpha } \right) = \left\\{ \begin{array}{l} \alpha \bmod 2\pi {\rm{, if }}\left( {\alpha \bmod 2\pi } \right) \in \left[ {0,\frac{\pi }{2}} \right] \\\\ \left( {\alpha \bmod 2\pi } \right) – 2\pi ,{\rm{ if }}\left( {\alpha \bmod 2\pi } \right) \in \left[ {\frac{{3\pi }}{2},2\pi } \right] \\\\ \pi – \left( {\alpha \bmod 2\pi } \right),{\rm{ else}} \end{array} \right. $$

Arccos of cos

$$ \arccos \left( {\cos \alpha } \right) = \left\\{ \begin{array}{l} \alpha \bmod 2\pi {\rm{, if }}\left( {\alpha \bmod 2\pi } \right) \in \left[ {0,\pi } \right] \\\\ \left( { – \alpha } \right)\bmod 2\pi {\rm{, else}} \end{array} \right. $$

Arccot of cot and arctan of tan

$$ \begin{array}{l} {\mathop{\rm arccot}\nolimits} \left( {\cot \alpha } \right) = \alpha \bmod \pi \\\\ \arctan \left( {\tan \alpha } \right) = \left\\{ \begin{array}{l} \alpha \bmod \pi {\rm{, if }}\left( {\alpha \bmod \pi } \right) \in \left[ {0,\frac{\pi }{2}} \right] \\\\ \left( {\alpha \bmod \pi } \right) – \pi {\rm{, else}} \end{array} \right. \end{array} $$

逆三角関数の関係（Relations between arcus functions）

$$ \begin{array}{l} \arccos a + \arcsin a = \frac{\pi }{2}\\\\ \arctan a + {\mathop{\rm arccot}\nolimits} a = \frac{\pi }{2} \end{array} $$

単位円の対称性（Unit circle symmetry）

X-axis

$$ \begin{array}{l} \sin x = – \sin \left( { – x} \right)\\\\ \cos x = \cos \left( { – x} \right)\\\\ \tan x = – \tan \left( { – x} \right) \end{array} $$

Y-axis

$$ \begin{array}{l} \sin x = \sin \left( {\pi – x} \right)\\\\ \cos x = – \cos \left( {\pi – x} \right)\\\\ \tan x = – \tan \left( {\pi – x} \right) \end{array} $$

座標原点（Coordinate origin）

$$ \begin{array}{l} \sin x = – \sin \left( {\pi + x} \right)\\\\ \cos x = – \cos \left( {\pi + x} \right)\\\\ \tan x = \tan \left( {\pi + x} \right) \end{array} $$

角の二等分線（Angle bisector）

$$ \begin{array}{l} \sin \left( {\frac{\pi }{2} – x} \right) = \cos x\\\\ \cos \left( {\frac{\pi }{2} – x} \right) = \sin x\\\\ \tan \left( {\frac{\pi }{2} – x} \right) = \cot x \end{array} $$

倍角と半角（Double and half angle）

倍角の公式（Double angle）

$$ \begin{array}{l} \sin \left( {2\alpha } \right) = 2\sin \alpha \cos \alpha \\\\ \cos \left( {2\alpha } \right) = 1 – 2{\sin ^2}\alpha \end{array} $$

タンジェントの場合。

$$ \begin{array}{l} \sin \left( {2\alpha } \right) = \frac{{2\tan \alpha }}{{1 + {{\tan }^2}\alpha }}\\\\ \cos \left( {2\alpha } \right) = \frac{{1 – {{\tan }^2}\alpha }}{{1 + {{\tan }^2}\alpha }}\\\\ \tan \left( {2\alpha } \right) = \frac{{2\tan \alpha }}{{1 – {{\tan }^2}\alpha }} \end{array} $$

半角の公式（Half angle）

$$ \begin{array}{l} {\sin ^2}\left( {\frac{\alpha }{2}} \right) = \frac{{1 – \cos \alpha }}{2}\\\\ {\cos ^2}\left( {\frac{\alpha }{2}} \right) = \frac{{1 + \cos \alpha }}{2}\\\\ \tan \left( {\frac{\alpha }{2}} \right) = \frac{{\sin \alpha }}{{1 + \cos \alpha }}{\rm{, if }}\alpha \in \left[ {0,\pi } \right]\\\\ = \frac{{\tan \alpha }}{{1 + \sqrt {1 + {{\tan }^2}\alpha } }}{\rm{, if }}\alpha \in \left] { – \frac{\pi }{2},\frac{\pi }{2}} \right[ \end{array} $$

三角関数の特殊値（Special values）

$$ \begin{array}{c|cccccccc} \begin{matrix}\text{Radian}\\\\ \text{Degree}\end{matrix} & \begin{matrix}0\\\\ 0^\circ\end{matrix} & \begin{matrix}\frac{\pi}{12}\\\\ 15^\circ\end{matrix} & \begin{matrix}\frac{\pi}{8}\\\\ 22.5^\circ\end{matrix} & \begin{matrix}\frac{\pi}{6}\\\\ 30^\circ\end{matrix} & \begin{matrix}\frac{\pi}{4}\\\\ 45^\circ\end{matrix} & \begin{matrix}\frac{\pi}{3}\\\\ 60^\circ\end{matrix} & \begin{matrix}\frac{5\pi}{12}\\\\ 75^\circ\end{matrix} & \begin{matrix}\frac{\pi}{2}\\\\ 90^\circ\end{matrix} \\\\ \hline \sin & 0 & \frac{ \sqrt{6} – \sqrt{2} } {4} & \frac{ \sqrt{2 – \sqrt{2}} } {2} & \frac{1}{2} & \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{2} & \frac{ \sqrt{6} + \sqrt{2} } {4} & 1 \\\\ \cos & 1 & \frac{\sqrt{6}+\sqrt{2}}{4} & \frac{ \sqrt{2 + \sqrt{2}} } {2} & \frac{\sqrt{3}}{2} & \frac{\sqrt{2}}{2} & \frac{1}{2} & \frac{ \sqrt{6} – \sqrt{2}} {4} & 0 \\\\ \tan & 0 & 2-\sqrt{3} & \sqrt{2} – 1 & \frac{\sqrt{3}}{3} & 1 & \sqrt{3} & 2+\sqrt{3} & \infty \\\\ \cot & \infty & 2+\sqrt{3} & \sqrt{2} + 1 & \sqrt{3} & 1 & \frac{\sqrt{3}}{3} & 2-\sqrt{3} & 0 \\\\ \sec & 1 & \sqrt{6} – \sqrt{2} & \sqrt{2} \sqrt{ 2 – \sqrt{2} } & \frac{2\sqrt{3}}{3} & \sqrt{2} & 2 & \sqrt{6}+\sqrt{2} & \infty \\\\ \csc & \infty & \sqrt{6}+\sqrt{2} & \sqrt{2} \sqrt{ 2 + \sqrt{2} } & 2 & \sqrt{2} & \frac{2\sqrt{3}}{3} & \sqrt{6} – \sqrt{2} & 1 \\\\ \end{array} $$

三角関数の微分（Differentiation）

$$ \begin{align} \frac{d}{dx}\sin(x) =& \cos(x)\\\\ \frac{d}{dx}\cos(x) =& -\sin(x)\\\\ \frac{d}{dx}\tan(x) =& \left(\frac{\sin(x)}{\cos(x)}\right)’ = \frac{\cos^2(x) + \sin^2(x)}{\cos^2(x)} = 1 + \tan^2(x) = \sec^2(x)\\\\ \frac{d}{dx}\cot(x) =& \left(\frac{\cos(x)}{\sin(x)}\right)’ = \frac{-\sin^2(x) – \cos^2(x)}{\sin^2(x)} = -(1+\cot^2(x)) = -\csc^2(x)\\\\ \frac{d}{dx}\sec(x) =& \left(\frac{1}{\cos(x)}\right)’ = \frac{\sin(x)}{\cos^2(x)} = \frac{1}{\cos(x)} \cdot \frac{\sin(x)}{\cos(x)} = \sec(x)\tan(x)\\\\ \frac{d}{dx}\csc(x) =& \left(\frac{1}{\sin(x)}\right)’ = -\frac{\cos(x)}{\sin^2(x)} = -\frac{1}{\sin(x)} \cdot \frac{\cos(x)}{\sin(x)} = -\csc(x)\cot(x)\\\\ \frac{d}{dx}\arcsin(x) =& \frac{1}{\sqrt{1-x^2}}\\\\ \frac{d}{dx}\arccos(x) =& \frac{-1}{\sqrt{1-x^2}}\\\\ \frac{d}{dx}\arctan(x) =& \frac{1}{1+x^2}\\\\ \frac{d}{dx}arccot(x) =& \frac{-1}{1+x^2} \\\\ \frac{d}{dx}arccsc(x) =& \frac{-1}{|x|\sqrt{x^2-1}} \end{align} $$

三角関数の積分（Integral）

不定積分（indefinite integral）

$$ \begin{align} &\int \sin^{2} x \, dx = \frac{x}{2} - \frac{1}{4} \sin 2x + C \\\\ &\int \cos^{2} x \, dx = \frac{x}{2} + \frac{1}{4} \sin 2x + C \\\\ &\int x \sin x \, dx = -x \cos x + \sin x + C \\\\ &\int x \cos x \, dx = x \sin x + \cos x + C \\\\ &\int x^{2} \sin x \, dx = -x^{2} \cos x + 2x \sin x + 2\cos x + C \\\\ &\int x^{2}\cos x \, dx = x^{2} \sin x + 2x \cos x - 2\sin x + C \end{align} $$

定積分（definite integral）

$$ \begin{align} &\int_{0}^{\textstyle \frac{\pi}{2}} \sin x \, dx = \int_{0}^{\textstyle \frac{\pi}{2}} \cos x \, dx = 1 \\\\ &\int_{0}^{\textstyle \frac{\pi}{2}} \sin^{2} x \, dx = \int_{0}^{\textstyle \frac{\pi}{2}} \cos^{2} x \, dx = \frac{\pi}{4} \\\\ &\int_{0}^{\textstyle \frac{\pi}{2}} \sin^{3} x \, dx = \int_{0}^{\textstyle \frac{\pi}{2}} \cos^{3} x \, dx = \frac{2}{3} \\\\ &\int_{0}^{\textstyle \frac{\pi}{2}} \sin^{4} x \, dx = \int_{0}^{\textstyle \frac{\pi}{2}} \cos^{4} x \, dx = \frac{3\pi}{16} \end{align} $$