初等関数 (elementary function)

\begin{eqnarray*}
\tan x &=& \frac{\sin x}{\cos x} \\
\\
\cot x &=& \frac{\cos x}{\sin x} = \frac{1}{\tan x} \\
\\
\sec x &=& \frac{1}{\cos x} \\
\\
\csc x &=& \frac{1}{\sin x} \\
\end{eqnarray*}

\tan x &=& \frac{\sin x}{\cos x}

\cot x &=& \frac{\cos x}{\sin x} = \frac{1}{\tan x}

\sec x &=& \frac{1}{\cos x}

\csc x &=& \frac{1}{\sin x}

\begin{eqnarray*}
\sin ^2 x + \cos ^2 &=& 1 \\
\\
1 + \tan ^2 x &=& \sec ^2 x
\end{eqnarray*}

\sin ^2 x + \cos ^2 &=& 1

1 + \tan ^2 x &=& \sec ^2 x

\begin{eqnarray*}
\sin (-x) &=& -\sin x \\
\\
\cos (-x) &=& \cos x \\
\\
\tan (-x) &=& -\tan x \\
\end{eqnarray*}

\sin (-x) &=& -\sin x

\cos (-x) &=& \cos x

\tan (-x) &=& -\tan x

\begin{eqnarray*}
\sin (x+y) &=& \sin x \cos y + \cos x \sin y \\
\\
\sin (x-y) &=& \sin x \cos y – \cos x \sin y \\
\\
\sin (x\pm y) &=& \sin x \cos y \pm \cos x \sin y \\
\end{eqnarray*}

\sin (x+y) &=& \sin x \cos y + \cos x \sin y

\sin (x-y) &=& \sin x \cos y – \cos x \sin y

\sin (x \pm y) &=& \sin x \cos y \pm \cos x \sin y

\begin{eqnarray*}
\cos (x+y) &=& \cos x \cos y – \sin x \sin y \\
\\
\cos (x-y) &=& \cos x \cos y + \sin x \sin y \\
\\
\cos (x \pm y) &=& \cos x \cos y \mp \sin x \sin y \\
\end{eqnarray*}

\cos (x+y) &=& \cos x \cos y – \sin x \sin y

\cos (x-y) &=& \cos x \cos y + \sin x \sin y

\cos (x \pm y) &=& \cos x \cos y \mp \sin x \sin y

\begin{eqnarray*}
\tan (x+y) &=& \frac{\tan x + \tan y}{1 – \tan x \tan y} \\
\\
\tan (x-y) &=& \frac{\tan x – \tan y}{1 + \tan x \tan y} \\
\\
\tan (x \pm y) &=& \frac{\tan x \pm \tan y}{1 \mp \tan x \tan y} \\
\end{eqnarray*}

\tan (x+y) &=& \frac{\tan x + \tan y}{1 – \tan x \tan y}

\tan (x-y) &=& \frac{\tan x – \tan y}{1 + \tan x \tan y}

\tan (x \pm y) &=& \frac{\tan x \pm \tan y}{1 \mp \tan x \tan y}

\begin{eqnarray*}
\sin x + \sin y &=& 2 \sin \frac{x+y}{2} \cos \frac{x-y}{2} \\
\\
\sin x – \sin y &=& 2 \cos \frac{x+y}{2} \sin \frac{x-y}{2} \\
\end{eqnarray*}

\sin x + \sin y &=& 2 \sin \frac{x+y}{2} \cos \frac{x-y}{2}

\sin x – \sin y &=& 2 \cos \frac{x+y}{2} \sin \frac{x-y}{2}

\begin{eqnarray*}
\cos x + \cos y &=& 2 \cos \frac{x+y}{2} \cos \frac{x-y}{2} \\
\\
\cos x – \cos y &=& -2 \sin \frac{x+y}{2} \sin \frac{x-y}{2} \\
\end{eqnarray*}

\cos x + \cos y &=& 2 \cos \frac{x+y}{2} \cos \frac{x-y}{2}

\cos x – \cos y &=& -2 \sin \frac{x+y}{2} \sin \frac{x-y}{2}

\begin{eqnarray*}
\sin ^2 \frac{x}{2} &=& \frac{1}{2} (1 – \cos x) \\
\\
\cos ^2 \frac{x}{2} &=& \frac{1}{2} (1 + \cos x) \\
\\
\tan ^2 \frac{x}{2} &=& \frac{1 – \cos x}{1 + \cos x} \\
\end{eqnarray*}

\sin ^2 \frac{x}{2} &=& \frac{1}{2} (1 – \cos x)

\cos ^2 \frac{x}{2} &=& \frac{1}{2} (1 + \cos x)

\tan ^2 \frac{x}{2} &=& \frac{1 – \cos x}{1 + \cos x}

\begin{eqnarray*}
\sin x \sin y &=& -\frac{1}{2} \{ \cos (x+y) – \cos (x-y) \} \\
\\
\cos x \cos y &=& \frac{1}{2} \{ \cos (x+y) + \cos (x-y) \} \\
\\
\sin x \cos y &=& \frac{1}{2} \{ \sin (x+y) + \sin (x-y) \} \\
\end{eqnarray*}

\sin x \sin y &=& -\frac{1}{2} \{ \cos (x+y) – \cos (x-y) \}

\cos x \cos y &=& \frac{1}{2} \{ \cos (x+y) + \cos (x-y) \}

\sin x \cos y &=& \frac{1}{2} \{ \sin (x+y) + \sin (x-y) \}

\begin{eqnarray*}
a^m \times a^n &=& a^{m+n} \\
\\
a^0 &=& 1 \\
\\
(a^m)^n &=& a^{mn} \\
\\
(ab)^m &=& a^n \cdot b^n \\
\\
a^{-1} &=& \frac{1}{a^n} \\
\\
a^{\frac{1}{n}} &=& ^n\sqrt{a}
\end{eqnarray*}

a^m \times a^n &=& a^{m+n}

a^0 &=& 1

(a^m)^n &=& a^{mn}

(ab)^m &=& a^n \cdot b^n

a^{-1} &=& \frac{1}{a^n}

a^{\frac{1}{n}} &=& ^n\sqrt{a}

\begin{eqnarray*}
\log _a xy &=& \log _a x + \log _a y \\
\\
\log _a x^u &=& u \log _a x \\
\\
\log _a x &=& \frac{\log _b x}{\log _b a} \\
\\
\log _a 1 &=& 0
\end{eqnarray*}

\begin{eqnarray*}
x &=& e^y \\
\\
\log _e x &=& \log_e e^y \\
\\
\log _e x &=& y \log_e e \\
\\
\log _e x &=& y
\end{eqnarray*}

x &=& e^y

\log _e x &=& \log_e e^y

\log _e x &=& y \log_e e

\log _e x &=& y

\begin{eqnarray*}
x &=& 10^y \\
\\
\log _{10} x &=& \log_{10} 10^y \\
\\
\log _{10} x &=& y \log_{10} 10 \\
\\
\log _{10} x &=& y
\end{eqnarray*}

x &=& 10^y

\log _{10} x &=& \log_{10} 10^y

\log _{10} x &=& y \log_{10} 10

\log _{10} x &=& y

\begin{eqnarray*}
x &=& a^y \\
\\
\log _a x &=& \log_a a^y \\
\\
\log _a x &=& y \log_a a \\
\\
\log _a x &=& y
\end{eqnarray*}

x &=& a^y

\log _a x &=& \log_a a^y

\log _a x &=& y \log_a a

\log _a x &=& y

$$
\log _e x = \frac{\log _{10} x}{\log _{10} e}
$$

\log _e x = \frac{\log _{10} x}{\log _{10} e}

\begin{eqnarray*}
\log (xy) &=& \log x + \log y \\
\\
\log \frac{x}{y} &=& \log x – \log y \\
\\
\log x^{\alpha} &=& \alpha \log x \\
\\
\frac{d}{dx} \log x &=& \frac{1}{x}
\end{eqnarray*}

\log (xy) &=& \log x + \log y

\log \frac{x}{y} &=& \log x – \log y

\log x^{\alpha} &=& \alpha \log x

\frac{d}{dx} \log x &=& \frac{1}{x}

\begin{eqnarray*}
\sinh x &=& \frac{e^x – e^{-x}}{2} \\
\\
\cosh x &=& \frac{e^x + e^{-x}}{2}
\end{eqnarray*}

\sinh x &=& \frac{e^x – e^{-x}}{2}

\cosh x &=& \frac{e^x + e^{-x}}{2}