三角関数と双曲線関数の２倍角と３倍角公式
三角関数の２倍角公式
(1)
\[ \sin2x=2\sin x\cos x \](2)
\begin{align*} \cos2x & =2\cos^{2}x-1\\ & =1-2\sin^{2}x\\ & =\cos^{2}x-\sin^{2}x \end{align*}(3)
\[ \tan2x=\frac{2\tan x}{1-\tan^{2}x} \](1)
\begin{align*} \sin2x & =\sin(x+x)\\ & =\sin x\cos x+\cos x\sin x\\ & =2\sin x\cos x \end{align*}(2)
\begin{align*} \cos2x & =\cos(x+x)\\ & =\cos x\cos x-\sin x\sin x\\ & =\cos^{2}x-\sin^{2}x\tag{*}\\ & =2\cos^{2}x-1\tag{*}\\ & =1-2\sin^{2}x\tag{*} \end{align*}(3)
\begin{align*} \tan2x & =\tan(x+x)\\ & =\frac{\tan x+\tan x}{1-\tan x\tan x}\\ & =\frac{2\tan x}{1-\tan^{2}x} \end{align*}双曲線関数の２倍角公式
(1)
\[ \sinh2x=2\sinh x\cosh x \](2)
\begin{align*} \cosh2x & =2\cosh^{2}x-1\\ & =1+2\sinh^{2}x\\ & =\cosh^{2}x+\sinh^{2}x \end{align*}(3)
\[ \tanh2x=\frac{2\tanh x}{1+\tanh^{2}x} \](1)
\begin{align*} \sinh2x & =-i\sin(2ix)\\ & =-2i\sin(ix)\cos(ix)\\ & =2\sinh x\cosh x \end{align*}(2)
\begin{align*} \cosh2x & =\cos(2ix)\\ & =2\cos^{2}(ix)-1\\ & =2\cosh^{2}x-1\tag{*}\\ & =1+2\sinh^{2}x\tag{*}\\ & =\cosh^{2}x+\sinh^{2}x\tag{*} \end{align*}(3)
\begin{align*} \tanh2x & =-i\tan(2ix)\\ & =-i\frac{2\tan(ix)}{1-\tan^{2}(ix)}\\ & =\frac{2\tanh x}{1+\tanh^{2}x} \end{align*}三角関数の３倍角公式
(1)
\[ \sin3x=3\sin x-4\sin^{3}x \](2)
\[ \cos3x=4\cos^{3}x-3\cos x \](3)
\[ \tan3x=\frac{3\tan x-\tan^{3}x}{1-3\tan^{2}x} \](1)
\begin{align*} \sin3x & =\sin(x+2x)\\ & =\sin x\cos2x+\cos x\sin2x\\ & =\sin x(1-2\sin^{2}x)+\cos x(2\sin x\cos x)\\ & =3\sin x-4\sin^{3}x \end{align*}(2)
\begin{align*} \cos3x & =\cos(x+2x)\\ & =\cos x\cos2x-\sin x\sin2x\\ & =\cos x(2\cos^{2}x-1)-\sin x(2\sin x\cos x)\\ & =4\cos^{3}x-3\cos x \end{align*}(3)
\begin{align*} \tan3x & =\frac{\sin3x}{\cos3x}\\ & =\frac{3\sin x-4\sin^{3}x}{4\cos^{3}x-3\cos x}\\ & =\frac{3\tan x\cos^{-2}x-4\tan^{3}x}{4-3\cos^{-2}x}\\ & =\frac{3\tan x(1+\tan^{2}x)-4\tan^{3}x}{4-3(1+\tan^{2}x)}\\ & =\frac{3\tan x-\tan^{3}x}{1-3\tan^{2}x} \end{align*}双曲線関数の３倍角公式
(1)
\[ \sinh3x=3\sinh x+4\sinh^{3}x \](2)
\[ \cosh3x=4\cosh^{3}x-3\cosh x \](3)
\[ \tanh3x=\frac{3\tanh x+\tanh^{3}x}{1+3\tanh^{2}x} \](1)
\begin{align*} \sinh3x & =-i\sin3ix\\ & =-i\left\{ 3\sin(ix)-4\sin^{3}(ix)\right\} \\ & =3\sinh x+4\sinh^{3}x \end{align*}(2)
\begin{align*} \cosh3x & =\cos(3ix)\\ & =4\cos^{3}(ix)-3\cos(ix)\\ & =4\cosh^{3}x--3\cosh x \end{align*}(3)
\begin{align*} \tanh3x & =-i\tan(3ix)\\ & =-i\frac{3\tan(ix)-\tan^{3}(ix)}{1-3\tan^{2}(ix)}\\ & =\frac{3\tanh x+\tanh^{3}x}{1+3\tanh^{2}x} \end{align*}
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| タイトル | 三角関数と双曲線関数の２倍角と３倍角公式 |
| URL | https://www.nomuramath.com/wj8itk5o/ |
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正接関数・双曲線正接関数の多重対数関数表示
\[
\tan^{\pm1}z=i^{\pm1}\left(1+2\Li_{0}\left(\mp e^{2iz}\right)\right)
\]
双曲線関数と三角関数の級数展開
\[
\tanh x=\sum_{k=1}^{\infty}\frac{2^{2k}\left(2^{2k}-1\right)B_{2k}}{(2k)!}x{}^{2k-1}
\]
逆三角関数と逆双曲線関数の積分表示
\[
\sin^{\bullet}x=\int_{0}^{x}\frac{1}{\sqrt{1-t^{2}}}dt
\]
三角関数と双曲線関数の積分
\[
\int f(\cos x,\sin x)dx=\int f\left(\frac{1-t^{2}}{1+t^{2}},\frac{2t}{1+t^{2}}\right)\frac{2}{1+t^{2}}dt\cnd{t=\tan\frac{x}{2}}
\]