3角関数・双曲線関数の総和
3角関数・双曲線関数の総和
3角関数・双曲線関数の総和について次が成り立つ。
正弦
\[ \sum_{k=m_{1}}^{m_{2}}\sin\left(ak+b\right)=\sin^{-1}\left(\frac{a}{2}\right)\sin\left(\left(m_{1}+m_{2}\right)\frac{a}{2}+b\right)\sin\left(\left(1+m_{2}-m_{1}\right)\frac{a}{2}\right) \]
余弦
\[ \sum_{k=m_{1}}^{m_{2}}\cos\left(ak+b\right)=\sin^{-1}\left(\frac{a}{2}\right)\cos\left(\left(m_{1}+m_{2}\right)\frac{a}{2}+b\right)\sin\left(\left(1+m_{2}-m_{1}\right)\frac{a}{2}\right) \]
\[ \sum_{k=1}^{2n}\cos\left(\frac{ak+b}{an}\pi\right)=0 \] この式で\(\cos\rightarrow\cosh\)にしても等号は成り立ちません。
正接
双曲線正弦関数
\[ \sum_{k=m_{1}}^{m_{2}}\sinh\left(ak+b\right)=\sinh^{-1}\left(\frac{a}{2}\right)\sinh\left(\left(m_{1}+m_{2}\right)\frac{a}{2}+b\right)\sinh\left(\left(1+m_{2}-m_{1}\right)\frac{a}{2}\right) \]
双曲線正弦関数
\[ \sum_{k=m_{1}}^{m_{2}}\cosh\left(ak+b\right)=\sinh^{-1}\left(\frac{a}{2}\right)\cosh\left(\left(m_{1}+m_{2}\right)\frac{a}{2}+b\right)\sinh\left(\left(1+m_{2}-m_{1}\right)\frac{a}{2}\right) \]
3角関数・双曲線関数の総和について次が成り立つ。
正弦
(1)
\(a\in\mathbb{Z}\setminus\left\{ 0\right\} ,b\in\mathbb{Z},m_{1},m_{2}\in\mathbb{Z}\)とする。\[ \sum_{k=m_{1}}^{m_{2}}\sin\left(ak+b\right)=\sin^{-1}\left(\frac{a}{2}\right)\sin\left(\left(m_{1}+m_{2}\right)\frac{a}{2}+b\right)\sin\left(\left(1+m_{2}-m_{1}\right)\frac{a}{2}\right) \]
(2)
\[ \sum_{k=1}^{n-1}\sin\left(\frac{k}{n}\pi\right)=\tan^{-1}\left(\frac{\pi}{2n}\right) \] この式で\(\sin\rightarrow\sinh,\tan\rightarrow\tanh\)にしても等号は成り立ちません。(3)
\[ \sum_{k=-n}^{n}\sin\left(\frac{k}{a}\pi\right)=0 \](4)
\[ \sum_{k=1}^{n-1}\sin\left(\frac{k}{2n-1}\pi\right)=\frac{1}{2}\tan^{-1}\left(\frac{\pi}{4n-2}\right) \] この式で\(\sin\rightarrow\sinh,\tan\rightarrow\tanh\)にしても等号は成り立ちません。(5)
\[ \sum_{k=1}^{n-1}\sin\left(\frac{2k}{2n-1}\pi\right)=\frac{1}{2}\tan^{-1}\left(\frac{\pi}{4n-2}\right) \] この式で\(\sin\rightarrow\sinh,\tan\rightarrow\tanh\)にしても等号は成り立ちません。(6)
\[ \sum_{k=1}^{n-1}\sin\left(\frac{2k-1}{2n-1}\pi\right)=\frac{1}{2}\tan^{-1}\left(\frac{\pi}{4n-2}\right) \] この式で\(\sin\rightarrow\sinh,\tan\rightarrow\tanh\)にしても等号は成り立ちません。(7)
\[ \sum_{k=1}^{n}\sin\left(\frac{2k-1}{2n}\pi\right)=\sin^{-1}\frac{\pi}{2n} \] この式で\(\sin\rightarrow\sinh\)にしても等号は成り立ちません。(8)
\[ \sum_{k=0}^{n}\sin\left(\frac{k}{2n}\pi\right)=\frac{1}{2}\left(\tan^{-1}\left(\frac{\pi}{4n}\right)+1\right) \] この式で\(\sin\rightarrow\sinh,\tan\rightarrow\tanh\)にしても等号は成り立ちません。余弦
(9)
\(a\in\mathbb{Z}\setminus\left\{ 0\right\} ,b\in\mathbb{Z},m_{1},m_{2}\in\mathbb{Z}\)とする。\[ \sum_{k=m_{1}}^{m_{2}}\cos\left(ak+b\right)=\sin^{-1}\left(\frac{a}{2}\right)\cos\left(\left(m_{1}+m_{2}\right)\frac{a}{2}+b\right)\sin\left(\left(1+m_{2}-m_{1}\right)\frac{a}{2}\right) \]
(10)
\[ \sum_{k=-n}^{n}\cos\left(\frac{k}{2n}\pi\right)=\tan^{-1}\left(\frac{\pi}{4n}\right) \] この式で\(\cos\rightarrow\cosh,\tan\rightarrow\tanh\)にしても等号は成り立ちません。(11)
\[ \sum_{k=-\left(n-1\right)}^{n-1}\cos\left(\frac{k}{2n-1}\pi\right)=\sin^{-1}\left(\frac{\pi}{4n-2}\right) \] この式で\(\cos\rightarrow\cosh,\sin\rightarrow\sinh\)にしても等号は成り立ちません。(12)
\begin{align*} \sum_{k=0}^{n}\cos\left(\frac{k}{n}\pi\right) & =0 \end{align*} この式で\(\cos\rightarrow\cosh\)にしても等号は成り立ちません。(13)
\[ \sum_{k=-n}^{n}\cos\left(\frac{k}{n}\pi\right)=-1 \] この式で\(\cos\rightarrow\cosh\)にしても等号は成り立ちません。(14)
\(a\in\mathbb{Z}\setminus\left\{ 0\right\} ,b\in\mathbb{Z}\)とする。\[ \sum_{k=1}^{2n}\cos\left(\frac{ak+b}{an}\pi\right)=0 \] この式で\(\cos\rightarrow\cosh\)にしても等号は成り立ちません。
(15)
\[ \sum_{k=0}^{n}\cos\left(\frac{k}{2n}\pi\right)=\frac{1}{2}\left(\tan^{-1}\left(\frac{\pi}{4n}\right)+1\right) \] この式で\(\cos\rightarrow\cosh,\tan\rightarrow\tanh\)にしても等号は成り立ちません。正接
(16)
\[ \sum_{k=0}^{2n+1}\tan\frac{k\pi}{2n+1}=0 \] この式で\(\tan\rightarrow\tanh\)にしても等号は成り立ちません。双曲線正弦関数
(17)
\(a\in\mathbb{Z}\setminus\left\{ 0\right\} ,b\in\mathbb{Z},m_{1},m_{2}\in\mathbb{Z}\)とする。\[ \sum_{k=m_{1}}^{m_{2}}\sinh\left(ak+b\right)=\sinh^{-1}\left(\frac{a}{2}\right)\sinh\left(\left(m_{1}+m_{2}\right)\frac{a}{2}+b\right)\sinh\left(\left(1+m_{2}-m_{1}\right)\frac{a}{2}\right) \]
(18)
\[ \sum_{k=-n}^{n}\sinh\left(\frac{k}{a}\pi\right)=0 \]双曲線正弦関数
(19)
\(a\in\mathbb{Z}\setminus\left\{ 0\right\} ,b\in\mathbb{Z},m_{1},m_{2}\in\mathbb{Z}\)とする。\[ \sum_{k=m_{1}}^{m_{2}}\cosh\left(ak+b\right)=\sinh^{-1}\left(\frac{a}{2}\right)\cosh\left(\left(m_{1}+m_{2}\right)\frac{a}{2}+b\right)\sinh\left(\left(1+m_{2}-m_{1}\right)\frac{a}{2}\right) \]
3角を双曲線関数に置き換えても等号が成り立つとは限りません。
例として、
\[ \sum_{k=1}^{n-1}\sinh\left(\frac{k}{n}\pi\right)\ne\tanh^{-1}\left(\frac{\pi}{2n}\right) \] の等号が成り立たないことを示す。
\(n=1\)を代入すると、
\begin{align*} \sum_{k=1}^{0}\sinh\left(\frac{k}{1}\pi\right) & =0\\ & \ne\tanh^{-1}\left(\frac{\pi}{2\cdot1}\right)\\ & =\tanh^{-1}\left(\frac{\pi}{2}\right) \end{align*} となるからです。
次に
\[ \sum_{k=0}^{2n+1}\tanh\frac{k\pi}{2n+1}=0 \] が成り立たないことを示す。
\(n=0\)を代入すると、
\begin{align*} \sum_{k=0}^{1}\tanh k\pi & =\tanh0+\tanh\pi\\ & =0+\frac{e^{\pi}+e^{-\pi}}{e^{\pi}-e^{-\pi}}\\ & >0 \end{align*} となり、\(\sum_{k=0}^{1}\tanh k\pi\ne0\)となるからです。
例として、
\[ \sum_{k=1}^{n-1}\sinh\left(\frac{k}{n}\pi\right)\ne\tanh^{-1}\left(\frac{\pi}{2n}\right) \] の等号が成り立たないことを示す。
\(n=1\)を代入すると、
\begin{align*} \sum_{k=1}^{0}\sinh\left(\frac{k}{1}\pi\right) & =0\\ & \ne\tanh^{-1}\left(\frac{\pi}{2\cdot1}\right)\\ & =\tanh^{-1}\left(\frac{\pi}{2}\right) \end{align*} となるからです。
次に
\[ \sum_{k=0}^{2n+1}\tanh\frac{k\pi}{2n+1}=0 \] が成り立たないことを示す。
\(n=0\)を代入すると、
\begin{align*} \sum_{k=0}^{1}\tanh k\pi & =\tanh0+\tanh\pi\\ & =0+\frac{e^{\pi}+e^{-\pi}}{e^{\pi}-e^{-\pi}}\\ & >0 \end{align*} となり、\(\sum_{k=0}^{1}\tanh k\pi\ne0\)となるからです。
(1)
\begin{align*} \sum_{k=m_{1}}^{m_{2}}\sin\left(ak+b\right) & =\sin^{-1}\left(\frac{a}{2}\right)\sin\left(\frac{a}{2}\right)\sum_{k=m_{1}}^{m_{2}}\sin\left(ak+b\right)\\ & =\sin^{-1}\left(\frac{a}{2}\right)\sum_{k=m_{1}}^{m_{2}}\sin\left(\frac{a}{2}\right)\sin\left(ak+b\right)\\ & =\sin^{-1}\left(\frac{a}{2}\right)\sum_{k=m_{1}}^{m_{2}}-\frac{1}{2}\left\{ \cos\left(\frac{a}{2}+\left(ak+b\right)\right)-\cos\left(\frac{a}{2}-\left(ak+b\right)\right)\right\} \cmt{\because\sin\alpha\sin\beta=-\frac{1}{2}\left(\cos\left(\alpha+\beta\right)-\cos\left(\alpha-\beta\right)\right)}\\ & =-\frac{1}{2}\sin^{-1}\left(\frac{a}{2}\right)\sum_{k=m_{1}}^{m_{2}}\left\{ \cos\left(\left(k+\frac{1}{2}\right)a+b\right)-\cos\left(-\left(k-\frac{1}{2}\right)a-b\right)\right\} \\ & =-\frac{1}{2}\sin^{-1}\left(\frac{a}{2}\right)\sum_{k=m_{1}}^{m_{2}}\left\{ \cos\left(\left(k+\frac{1}{2}\right)a+b\right)-\cos\left(\left(k-\frac{1}{2}\right)a+b\right)\right\} \\ & =-\frac{1}{2}\sin^{-1}\left(\frac{a}{2}\right)\left\{ \cos\left(\left(m_{2}+\frac{1}{2}\right)a+b\right)-\cos\left(\left(m_{1}-\frac{1}{2}\right)a+b\right)\right\} \\ & =-\frac{1}{2}\sin^{-1}\left(\frac{a}{2}\right)\left\{ -2\sin\left(\frac{\left(m_{2}+\frac{1}{2}\right)a+b+\left(m_{1}-\frac{1}{2}\right)a+b}{2}\right)\sin\left(\frac{\left(m_{2}+\frac{1}{2}\right)a+b-\left(\left(m_{1}-\frac{1}{2}\right)a+b\right)}{2}\right)\right\} \cmt{\because\cos\alpha-\cos\beta=-2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}}\\ & =\sin^{-1}\left(\frac{a}{2}\right)\sin\left(\left(m_{1}+m_{2}\right)\frac{a}{2}+b\right)\sin\left(\left(1+m_{2}-m_{1}\right)\frac{a}{2}\right) \end{align*}(2)
\begin{align*} \sum_{k=1}^{n-1}\sin\left(\frac{k}{n}\pi\right) & =\sum_{k=1}^{n-1}\sin\left(\frac{\pi}{n}k\right)\\ & =\sin^{-1}\left(\frac{1}{2}\cdot\frac{\pi}{n}\right)\sin\left(\left(1+n-1\right)\frac{1}{2}\cdot\frac{\pi}{n}+0\right)\sin\left(\left(1+n-1-1\right)\frac{1}{2}\cdot\frac{\pi}{n}\right)\cmt{\because\sum_{k=m_{1}}^{m_{2}}\sin\left(ak+b\right)=\sin^{-1}\left(\frac{a}{2}\right)\sin\left(\left(m_{1}+m_{2}\right)\frac{a}{2}+b\right)\sin\left(\left(1+m_{2}-m_{1}\right)\frac{a}{2}\right)}\\ & =\sin^{-1}\left(\frac{\pi}{2n}\right)\sin\left(\frac{\pi}{2}\right)\sin\left(\left(n-1\right)\frac{\pi}{2n}\right)\\ & =\sin^{-1}\left(\frac{\pi}{2n}\right)\sin\left(\frac{\pi}{2}-\frac{\pi}{2n}\right)\\ & =\sin^{-1}\left(\frac{\pi}{2n}\right)\cos\left(\frac{\pi}{2n}\right)\\ & =\tan^{-1}\left(\frac{\pi}{2n}\right) \end{align*}(2)-2
\begin{align*} \sum_{k=1}^{n-1}\sin\left(\frac{k}{n}\pi\right) & =\sin^{-1}\left(\frac{\pi}{n}\right)\sin\left(\frac{\pi}{n}\right)\sum_{k=1}^{n-1}\sin\left(\frac{k}{n}\pi\right)\\ & =\sin^{-1}\left(\frac{\pi}{n}\right)\sum_{k=1}^{n-1}\sin\left(\frac{k}{n}\pi\right)\sin\left(\frac{\pi}{n}\right)\\ & =\sin^{-1}\left(\frac{\pi}{n}\right)\sum_{k=1}^{n-1}-\frac{1}{2}\left\{ \cos\left(\frac{k}{n}\pi+\frac{\pi}{n}\right)-\cos\left(\frac{k}{n}\pi-\frac{\pi}{n}\right)\right\} \\ & =-\frac{1}{2}\sin^{-1}\left(\frac{\pi}{n}\right)\sum_{k=1}^{n-1}\left\{ \cos\left(\frac{k+1}{n}\pi\right)-\cos\left(\frac{k-1}{n}\pi\right)\right\} \\ & =-\frac{1}{2}\sin^{-1}\left(\frac{\pi}{n}\right)\sum_{k=1}^{n-1}\left\{ \cos\left(\frac{\left(k-1\right)+2}{n}\pi\right)-\cos\left(\frac{k-1}{n}\pi\right)\right\} \\ & =-\frac{1}{2}\sin^{-1}\left(\frac{\pi}{n}\right)\left\{ -\cos\left(\frac{1-1}{n}\pi\right)-\cos\left(\frac{2-1}{n}\pi\right)+\cos\left(\frac{\left(\left(n-2\right)-1\right)+2}{n}\pi\right)+\cos\left(\frac{\left(\left(n-1\right)-1\right)+2}{n}\pi\right)\right\} \\ & =-\frac{1}{2}\sin^{-1}\left(\frac{\pi}{n}\right)\left\{ -\cos\left(0\right)-\cos\left(\frac{\pi}{n}\right)+\cos\left(\frac{n-1}{n}\pi\right)+\cos\left(\frac{n}{n}\pi\right)\right\} \\ & =-\frac{1}{2}\sin^{-1}\left(\frac{\pi}{n}\right)\left\{ -1-\cos\left(\frac{\pi}{n}\right)+\cos\left(\pi-\frac{1}{n}\pi\right)+\cos\left(\pi\right)\right\} \\ & =-\frac{1}{2}\sin^{-1}\left(\frac{\pi}{n}\right)\left\{ -1-\cos\left(\frac{\pi}{n}\right)-\cos\left(\frac{\pi}{n}\right)-1\right\} \\ & =\sin^{-1}\left(\frac{\pi}{n}\right)\left\{ 1+\cos\left(\frac{\pi}{n}\right)\right\} \\ & =\sin^{-1}\left(\frac{\pi}{n}\right)\left(2\cos^{2}\frac{\pi}{2n}\right)\cmt{\because\cos^{2}\frac{\alpha}{2}=\frac{1+\cos\alpha}{2}}\\ & =\frac{2\cos^{2}\frac{\pi}{2n}}{\sin\left(\frac{\pi}{n}\right)}\\ & =\frac{2\cos^{2}\frac{\pi}{2n}}{2\sin\left(\frac{\pi}{2n}\right)\cos\left(\frac{\pi}{2n}\right)}\cmt{\because\sin2\alpha=2\sin\alpha\cos\alpha}\\ & =\frac{\cos\frac{\pi}{2n}}{\sin\left(\frac{\pi}{2n}\right)}\\ & =\tan^{-1}\left(\frac{\pi}{2n}\right) \end{align*}(3)
\begin{align*} \sum_{k=-n}^{n}\sin\left(\frac{k}{a}\pi\right) & =\frac{1}{2}\left\{ \sum_{k=-n}^{n}\sin\left(\frac{k}{a}\pi\right)+\sum_{k=-n}^{n}\sin\left(-\frac{j}{a}\pi\right)\right\} \cmt{k=-j}\\ & =\frac{1}{2}\left\{ \sum_{k=-n}^{n}\sin\left(\frac{k}{a}\pi\right)-\sum_{k=-n}^{n}\sin\left(\frac{k}{a}\pi\right)\right\} \\ & =\frac{1}{2}\sum_{k=-n}^{n}\left(\sin\left(\frac{k}{a}\pi\right)-\sin\left(\frac{k}{a}\pi\right)\right)\\ & =0 \end{align*}(4)
\begin{align*} \sum_{k=1}^{n-1}\sin\left(\frac{k}{2n-1}\pi\right) & =\sum_{k=1}^{2n-1}\sin\left(\frac{k}{2n-1}\pi\right)-\sum_{k=n}^{2n-1}\sin\left(\frac{k}{2n-1}\pi\right)\\ & =\sum_{k=1}^{2n-1}\sin\left(\frac{k}{2n-1}\pi\right)-\sum_{k=0}^{n-1}\sin\left(\frac{k+n}{2n-1}\pi\right)\\ & =\sum_{k=1}^{2n-1}\sin\left(\frac{k}{2n-1}\pi\right)-\sum_{k=0}^{n-1}\sin\left(\frac{\left(n-1-k\right)+n}{2n-1}\pi\right)\\ & =\sum_{k=1}^{2n-1}\sin\left(\frac{k}{2n-1}\pi\right)-\sum_{k=0}^{n-1}\sin\left(\frac{2n-1-k}{2n-1}\pi\right)\\ & =\sum_{k=1}^{2n-1}\sin\left(\frac{k}{2n-1}\pi\right)-\sum_{k=0}^{n-1}\sin\left(\pi-\frac{k}{2n-1}\pi\right)\\ & =\sum_{k=1}^{2n-1}\sin\left(\frac{k}{2n-1}\pi\right)-\sum_{k=0}^{n-1}\sin\left(\frac{k}{2n-1}\pi\right)\\ & =\frac{1}{2}\sum_{k=1}^{2n-1}\sin\left(\frac{k}{2n-1}\pi\right)\\ & =\frac{1}{2}\sum_{k=1}^{2n-1}\sin\left(\frac{k}{2\left(2n-1\right)}\pi\right)\\ & =\frac{1}{2}\tan^{-1}\left(\frac{\pi}{4n-2}\right) \end{align*}(5)
\begin{align*} \sum_{k=1}^{n-1}\sin\left(\frac{2k}{2n-1}\pi\right) & =\sum_{k=1}^{2n-1}\sin\left(\frac{k}{2n-1}\pi\right)-\sum_{k=1}^{n}\sin\left(\frac{2k-1}{2n-1}\pi\right)\\ & =\sum_{k=1}^{2n-1}\sin\left(\frac{k}{2n-1}\pi\right)-\sum_{k=1}^{n}\sin\left(\frac{2\left(n+1-k\right)-1}{2n-1}\pi\right)\\ & =\sum_{k=1}^{2n-1}\sin\left(\frac{k}{2n-1}\pi\right)-\sum_{k=1}^{n}\sin\left(\pi-\frac{2\left(k-1\right)}{2n-1}\pi\right)\\ & =\sum_{k=1}^{2n-1}\sin\left(\frac{k}{2n-1}\pi\right)-\sum_{k=1}^{n}\sin\left(\frac{2\left(k-1\right)}{2n-1}\pi\right)\\ & =\sum_{k=1}^{2n-1}\sin\left(\frac{k}{2n-1}\pi\right)-\sum_{k=0}^{n-1}\sin\left(\frac{2k}{2n-1}\pi\right)\\ & =\frac{1}{2}\sum_{k=1}^{2n-1}\sin\left(\frac{k}{2n-1}\pi\right)\\ & =\frac{1}{2}\tan^{-1}\left(\frac{\pi}{2\left(2n-1\right)}\right)\\ & =\frac{1}{2}\tan^{-1}\left(\frac{\pi}{4n-2}\right) \end{align*}(6)
\begin{align*} \sum_{k=1}^{n-1}\sin\left(\frac{2k-1}{2n-1}\pi\right) & =\sum_{k=1}^{2n-1}\sin\left(\frac{k}{2n-1}\pi\right)-\sum_{k=1}^{n-1}\sin\left(\frac{2k}{2n-1}\pi\right)\\ & =\tan^{-1}\left(\frac{\pi}{2\left(2n-1\right)}\right)-\frac{1}{2}\tan^{-1}\left(\frac{\pi}{2\left(2n-1\right)}\right)\\ & =\frac{1}{2}\tan^{-1}\left(\frac{\pi}{4n-2}\right) \end{align*}(7)
\begin{align*} \sum_{k=1}^{n}\sin\left(\frac{2k-1}{2n}\pi\right) & =\sum_{k=1}^{2n}\sin\left(\frac{k}{2n}\pi\right)-\sum_{k=1}^{n}\sin\left(\frac{2k}{2n}\pi\right)\\ & =\sum_{k=1}^{2n}\sin\left(\frac{k}{2n}\pi\right)-\sum_{k=1}^{n}\sin\left(\frac{k}{n}\pi\right)\\ & =\tan^{-1}\left(\frac{\pi}{4n}\right)-\tan^{-1}\left(\frac{\pi}{2n}\right)\\ & =\frac{-1-\tan^{-1}\frac{\pi}{4n}\tan^{-1}\frac{\pi}{2n}}{\tan^{-1}\left(\frac{\pi}{4n}-\frac{\pi}{2n}\right)}\cmt{\because\tan^{-1}\left(x\pm y\right)=\frac{-1\pm\tan^{-1}x\tan^{-1}y}{\tan^{-1}x\pm\tan^{-1}y}}\\ & =\frac{-1-\tan^{-1}\frac{\pi}{4n}\tan^{-1}\frac{\pi}{2n}}{\tan^{-1}\left(-\frac{\pi}{4n}\right)}\\ & =\frac{-1-\tan^{-1}\frac{\pi}{4n}\tan^{-1}\frac{\pi}{2n}}{-\tan^{-1}\frac{\pi}{4n}}\\ & =\frac{1+\tan^{-1}\frac{\pi}{4n}\tan^{-1}\frac{\pi}{2n}}{\tan^{-1}\frac{\pi}{4n}}\cdot\frac{\sin\frac{\pi}{4n}\cdot\sin\frac{\pi}{2n}}{\sin\frac{\pi}{4n}\cdot\sin\frac{\pi}{2n}}\\ & =\frac{\sin\frac{\pi}{4n}\cdot\sin\frac{\pi}{2n}+\cos\frac{\pi}{4n}\cos\frac{\pi}{2n}}{\sin\frac{\pi}{2n}\cos\frac{\pi}{4n}}\\ & =\frac{\cos\left(\frac{\pi}{4n}-\frac{\pi}{2n}\right)}{\sin\frac{\pi}{2n}\cos\frac{\pi}{4n}}\\ & =\frac{\cos\left(-\frac{\pi}{4n}\right)}{\sin\frac{\pi}{2n}\cos\frac{\pi}{4n}}\\ & =\frac{\cos\frac{\pi}{4n}}{\sin\frac{\pi}{2n}\cos\frac{\pi}{4n}}\\ & =\sin^{-1}\frac{\pi}{2n} \end{align*}(8)
\begin{align*} \sum_{k=1}^{n}\sin\left(\frac{k}{2n}\pi\right) & =\sum_{k=1}^{2n}\sin\left(\frac{k}{2n}\pi\right)-\sum_{k=n+1}^{2n}\sin\left(\frac{k}{2n}\pi\right)\\ & =\tan^{-1}\left(\frac{\pi}{2\cdot2n}\right)-\sum_{k=1}^{n}\sin\left(\frac{k+n}{2n}\pi\right)\cmt{k\rightarrow k+n}\\ & =\tan^{-1}\left(\frac{\pi}{4n}\right)-\sum_{k=0}^{n-1}\sin\left(\frac{\left(n-k\right)+n}{2n}\pi\right)\cmt{k\rightarrow n-k}\\ & =\tan^{-1}\left(\frac{\pi}{4n}\right)-\sum_{k=0}^{n-1}\sin\left(\pi-\frac{k}{2n}\pi\right)\\ & =\tan^{-1}\left(\frac{\pi}{4n}\right)-\sum_{k=0}^{n-1}\sin\left(\frac{k}{2n}\pi\right)\\ & =\tan^{-1}\left(\frac{\pi}{4n}\right)-\sum_{k=0}^{n}\sin\left(\frac{k}{2n}\pi\right)+\sin\left(\frac{n}{2n}\pi\right)\\ & =\tan^{-1}\left(\frac{\pi}{4n}\right)+\sum_{k=0}^{n}\sin\left(\frac{k}{2n}\pi\right)+\sin\left(\frac{\pi}{2}\right)\\ & =\tan^{-1}\left(\frac{\pi}{4n}\right)+\sum_{k=0}^{n}\sin\left(\frac{k}{2n}\pi\right)+1\\ & =\frac{1}{2}\left(\tan^{-1}\left(\frac{\pi}{4n}\right)+1\right) \end{align*}(9)
\begin{align*} \sum_{k=m_{1}}^{m_{2}}\cos\left(ak+b\right) & =\sum_{k=m_{1}}^{m_{2}}\sin\left(ak+\left(b+\frac{\pi}{2}\right)\right)\\ & =\sin^{-1}\left(\frac{a}{2}\right)\sin\left(\left(m_{1}+m_{2}\right)\frac{a}{2}+\left(b+\frac{\pi}{2}\right)\right)\sin\left(\left(1+m_{2}-m_{1}\right)\frac{a}{2}\right)\cmt{\because\sum_{k=m_{1}}^{m_{2}}\sin\left(ak+b\right)=\sin^{-1}\left(\frac{a}{2}\right)\sin\left(\left(m_{1}+m_{2}\right)\frac{a}{2}+b\right)\sin\left(\left(1+m_{2}-m_{1}\right)\frac{a}{2}\right)}\\ & =\sin^{-1}\left(\frac{a}{2}\right)\cos\left(\left(m_{1}+m_{2}\right)\frac{a}{2}+b\right)\sin\left(\left(1+m_{2}-m_{1}\right)\frac{a}{2}\right) \end{align*}(9)-2
\begin{align*} \sum_{k=m_{1}}^{m_{2}}\cos\left(ak+b\right) & =\sin^{-1}\left(\frac{a}{2}\right)\sin\left(\frac{a}{2}\right)\sum_{k=m_{1}}^{m_{2}}\cos\left(ak+b\right)\\ & =\sin^{-1}\left(\frac{a}{2}\right)\sum_{k=m_{1}}^{m_{2}}\sin\left(\frac{a}{2}\right)\cos\left(ak+b\right)\\ & =\sin^{-1}\left(\frac{a}{2}\right)\sum_{k=m_{1}}^{m_{2}}\frac{1}{2}\left\{ \sin\left(\frac{a}{2}+\left(ak+b\right)\right)+\sin\left(\frac{a}{2}-\left(ak+b\right)\right)\right\} \cmt{\because\sin\alpha\cos\beta=\frac{1}{2}\left(\sin\left(\alpha+\beta\right)+\sin\left(\alpha-\beta\right)\right)}\\ & =\frac{1}{2}\sin^{-1}\left(\frac{a}{2}\right)\sum_{k=m_{1}}^{m_{2}}\left\{ \sin\left(\left(k+\frac{1}{2}\right)a+b\right)+\sin\left(-\left(k-\frac{1}{2}\right)a-b\right)\right\} \\ & =\frac{1}{2}\sin^{-1}\left(\frac{a}{2}\right)\sum_{k=m_{1}}^{m_{2}}\left\{ \sin\left(\left(k+\frac{1}{2}\right)a+b\right)-\sin\left(\left(k-\frac{1}{2}\right)a+b\right)\right\} \\ & =\frac{1}{2}\sin^{-1}\left(\frac{a}{2}\right)\left\{ \sin\left(\left(m_{2}+\frac{1}{2}\right)a+b\right)-\sin\left(\left(m_{1}-\frac{1}{2}\right)a+b\right)\right\} \\ & =\frac{1}{2}\sin^{-1}\left(\frac{a}{2}\right)\left\{ 2\cos\left(\frac{\left(m_{2}+\frac{1}{2}\right)a+b+\left(m_{1}-\frac{1}{2}\right)a+b}{2}\right)\sin\left(\frac{\left(m_{2}+\frac{1}{2}\right)a+b-\left(\left(m_{1}-\frac{1}{2}\right)a+b\right)}{2}\right)\right\} \cmt{\because\sin\alpha-\sin\beta=2\cos\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}}\\ & =\sin^{-1}\left(\frac{a}{2}\right)\cos\left(\left(m_{1}+m_{2}\right)\frac{a}{2}+b\right)\sin\left(\left(1+m_{2}-m_{1}\right)\frac{a}{2}\right) \end{align*}(10)
\begin{align*} \sum_{k=-n}^{n}\cos\left(\frac{k}{2n}\pi\right) & =\sum_{k=0}^{2n}\cos\left(\frac{k-n}{2n}\pi\right)\\ & =\sum_{k=0}^{2n}\cos\left(\frac{k}{2n}\pi+\frac{\pi}{2}\right)\\ & =\sum_{k=0}^{2n}\sin\left(\frac{k}{2n}\pi\right)\\ & =\tan^{-1}\left(\frac{\pi}{2\cdot2n}\right)\\ & =\tan^{-1}\left(\frac{\pi}{4n}\right) \end{align*}(11)
\begin{align*} \sum_{k=-\left(n-1\right)}^{n-1}\cos\left(\frac{k}{2n-1}\pi\right) & =\sum_{k=-\left(n-1\right)}^{n-1}\cos\left(\frac{\pi}{2n-1}k\right)\\ & =\sin^{-1}\left(\frac{1}{2}\cdot\frac{\pi}{2n-1}\right)\cos\left(\left(-\left(n-1\right)+\left(n-1\right)\right)\frac{1}{2}\cdot\frac{\pi}{2n-1}+0\right)\sin\left(\left(1+\left(n-1\right)+\left(n-1\right)\right)\frac{1}{2}\cdot\frac{\pi}{2n-1}\right)\cmt{\because\sum_{k=m_{1}}^{m_{2}}\cos\left(ak+b\right)=\sin^{-1}\left(\frac{a}{2}\right)\cos\left(\left(m_{1}+m_{2}\right)\frac{a}{2}+b\right)\sin\left(\left(1+m_{2}-m_{1}\right)\frac{a}{2}\right)}\\ & =\sin^{-1}\left(\frac{\pi}{2\left(2n-1\right)}\right)\cos\left(0\right)\sin\left(\left(2n-1\right)\frac{1}{2}\cdot\frac{\pi}{2n-1}\right)\\ & =\sin^{-1}\left(\frac{\pi}{4n-2}\right)\sin\left(\frac{\pi}{2}\right)\\ & =\sin^{-1}\left(\frac{\pi}{4n-2}\right) \end{align*}(11)-2
\begin{align*} \sum_{k=-\left(n-1\right)}^{n-1}\cos\left(\frac{k}{2n-1}\pi\right) & =\sin^{-1}\left(\frac{\pi}{4n-2}\right)\sin\left(\frac{\pi}{4n-2}\right)\sum_{k=-\left(n-1\right)}^{n-1}\cos\left(\frac{k}{2n-1}\pi\right)\\ & =\sin^{-1}\left(\frac{\pi}{4n-2}\right)\sum_{k=-\left(n-1\right)}^{n-1}\sin\left(\frac{\pi}{4n-2}\right)\cos\left(\frac{k}{2n-1}\pi\right)\\ & =\sin^{-1}\left(\frac{\pi}{4n-2}\right)\sum_{k=-\left(n-1\right)}^{n-1}\frac{1}{2}\left\{ \sin\left(\frac{\pi}{4n-2}+\frac{k}{2n-1}\pi\right)+\sin\left(\frac{\pi}{4n-2}-\frac{k}{2n-1}\pi\right)\right\} \cmt{\because\sin\alpha\cos\beta=\frac{1}{2}\left\{ \sin\left(\alpha+\beta\right)+\sin\left(\alpha-\beta\right)\right\} }\\ & =\sin^{-1}\left(\frac{\pi}{4n-2}\right)\sum_{k=-\left(n-1\right)}^{n-1}\frac{1}{2}\left\{ \sin\left(\frac{2k+1}{4n-2}\pi\right)+\sin\left(-\frac{2k-1}{4n-2}\pi\right)\right\} \\ & =\sin^{-1}\left(\frac{\pi}{4n-2}\right)\sum_{k=-\left(n-1\right)}^{n-1}\frac{1}{2}\left\{ \sin\left(\frac{2k+1}{4n-2}\pi\right)-\sin\left(\frac{2k-1}{4n-2}\pi\right)\right\} \\ & =\sin^{-1}\left(\frac{\pi}{4n-2}\right)\cdot\frac{1}{2}\left\{ \sin\left(\frac{2\left(n-1\right)+1}{4n-2}\pi\right)-\sin\left(\frac{-2\left(n-1\right)-1}{4n-2}\pi\right)\right\} \\ & =\sin^{-1}\left(\frac{\pi}{4n-2}\right)\cdot\frac{1}{2}\left\{ \sin\left(\frac{2n-1}{4n-2}\pi\right)-\sin\left(-\frac{2n-1}{4n-2}\pi\right)\right\} \\ & =\sin^{-1}\left(\frac{\pi}{4n-2}\right)\cdot\frac{1}{2}\left\{ \sin\left(\frac{2n-1}{2\left(2n-1\right)}\pi\right)+\sin\left(\frac{2n-1}{2\left(2n-1\right)}\pi\right)\right\} \\ & =\sin^{-1}\left(\frac{\pi}{4n-2}\right)\cdot\frac{1}{2}\left\{ \sin\left(\frac{\pi}{2}\right)+\sin\left(\frac{\pi}{2}\right)\right\} \\ & =\sin^{-1}\left(\frac{\pi}{4n-2}\right) \end{align*}(12)
\begin{align*} \sum_{k=0}^{n}\cos\left(\frac{k}{n}\pi\right) & =\frac{1}{2}\sum_{k=0}^{n}\left(\cos\left(\frac{k}{n}\pi\right)+\cos\left(\frac{n-k}{n}\pi\right)\right)\\ & =\frac{1}{2}\sum_{k=0}^{n}\left(\cos\left(\frac{k}{n}\pi\right)+\cos\left(\pi-\frac{k}{n}\pi\right)\right)\\ & =\frac{1}{2}\sum_{k=0}^{n}\left(\cos\left(\frac{k}{n}\pi\right)-\cos\left(\frac{k}{n}\pi\right)\right)\\ & =0 \end{align*}(13)
\begin{align*} \sum_{k=-n}^{n}\cos\left(\frac{k}{n}\pi\right) & =\sum_{k=-n}^{0}\cos\left(\frac{k}{n}\pi\right)+\sum_{k=1}^{n}\cos\left(\frac{k}{n}\pi\right)\\ & =\sum_{k=0}^{n}\cos\left(-\frac{k}{n}\pi\right)+\sum_{k=0}^{n}\cos\left(\frac{k}{n}\pi\right)-\cos\left(\frac{0}{n}\pi\right)\\ & =\sum_{k=0}^{n}\cos\left(\frac{k}{n}\pi\right)+\sum_{k=0}^{n}\cos\left(\frac{k}{n}\pi\right)-1\\ & =0+0-1\\ & =-1 \end{align*}(14)
\begin{align*} \sum_{k=1}^{2n}\cos\left(\frac{ak+b}{an}\pi\right) & =\sum_{k=1}^{n}\cos\left(\frac{ak+b}{an}\pi\right)+\sum_{k=n+1}^{2n}\cos\left(\frac{ak+b}{an}\pi\right)\\ & =\sum_{k=1}^{n}\cos\left(\frac{ak+b}{an}\pi\right)+\sum_{k=1}^{n}\cos\left(\frac{a\left(k+n\right)+b}{an}\pi\right)\\ & =\sum_{k=1}^{n}\cos\left(\frac{ak+b}{an}\pi\right)+\sum_{k=1}^{n}\cos\left(\pi+\frac{ak+b}{an}\pi\right)\\ & =\sum_{k=1}^{n}\cos\left(\frac{ak+b}{an}\pi\right)-\sum_{k=1}^{n}\cos\left(\frac{ak+b}{an}\pi\right)\\ & =0 \end{align*}(15)
\begin{align*} \sum_{k=0}^{n}\cos\left(\frac{k}{2n}\pi\right) & =\sum_{k=-n}^{n}\cos\left(\frac{k}{2n}\pi\right)-\sum_{k=-n}^{-1}\cos\left(\frac{k}{2n}\pi\right)\\ & =\tan^{-1}\left(\frac{\pi}{4n}\right)-\sum_{k=1}^{n}\cos\left(\frac{-k}{2n}\pi\right)\\ & =\tan^{-1}\left(\frac{\pi}{4n}\right)-\sum_{k=1}^{n}\cos\left(\frac{k}{2n}\pi\right)\\ & =\tan^{-1}\left(\frac{\pi}{4n}\right)-\sum_{k=0}^{n}\cos\left(\frac{k}{2n}\pi\right)+1\\ & =\tan^{-1}\left(\frac{\pi}{4n}\right)-\sum_{k=0}^{n}\cos\left(\frac{k}{2n}\pi\right)+1\\ & =\frac{1}{2}\left(\tan^{-1}\left(\frac{\pi}{4n}\right)+1\right) \end{align*}(16)
\begin{align*} \sum_{k=0}^{2n+1}\tan\frac{k\pi}{2n+1} & =\frac{1}{2}\sum_{k=0}^{2n+1}\left(\tan\frac{k\pi}{2n+1}+\tan\frac{\left(2n+1-k\right)\pi}{2n+1}\right)\\ & =\frac{1}{2}\sum_{k=0}^{2n+1}\left(\tan\frac{k\pi}{2n+1}+\tan\left(\pi-\frac{k\pi}{2n+1}\right)\right)\\ & =\frac{1}{2}\sum_{k=0}^{2n+1}\left(\tan\frac{k\pi}{2n+1}-\tan\frac{k\pi}{2n+1}\right)\\ & =\frac{1}{2}\sum_{k=0}^{2n+1}0\\ & =0 \end{align*}(17)
\begin{align*} \sum_{k=m_{1}}^{m_{2}}\sinh\left(ak+b\right) & =-i\sum_{k=m_{1}}^{m_{2}}\sin\left(i\left(ak+b\right)\right)\\ & =-i\sum_{k=m_{1}}^{m_{2}}\sin\left(iak+ib\right)\\ & =-i\left\{ \sin^{-1}\left(\frac{ia}{2}\right)\sin\left(\left(m_{1}+m_{2}\right)\frac{ia}{2}+ib\right)\sin\left(\left(1+m_{2}-m_{1}\right)\frac{ia}{2}\right)\right\} \cmt{\because\sum_{k=m_{1}}^{m_{2}}\sin\left(ak+b\right)=\sin^{-1}\left(\frac{a}{2}\right)\sin\left(\left(m_{1}+m_{2}\right)\frac{a}{2}+b\right)\sin\left(\left(1+m_{2}-m_{1}\right)\frac{a}{2}\right)}\\ & =-i\left\{ i^{-1}\sinh^{-1}\left(\frac{a}{2}\right)i\sinh\left(\left(m_{1}+m_{2}\right)\frac{a}{2}+b\right)i\sinh\left(\left(1+m_{2}-m_{1}\right)\frac{a}{2}\right)\right\} \\ & =\sinh^{-1}\left(\frac{a}{2}\right)\sinh\left(\left(m_{1}+m_{2}\right)\frac{a}{2}+b\right)\sinh\left(\left(1+m_{2}-m_{1}\right)\frac{a}{2}\right) \end{align*}(18)
\begin{align*} \sum_{k=-n}^{n}\sinh\left(\frac{k}{a}\pi\right) & =-i\sum_{k=-n}^{n}\sin\left(i\frac{k}{a}\pi\right)\\ & =-i\sum_{k=-n}^{n}\sin\left(\frac{k}{-ia}\pi\right)\\ & =-i\cdot0\cmt{\because\sum_{k=-n}^{n}\sinh\left(\frac{k}{a}\pi\right)=0}\\ & =0 \end{align*}(19)
\begin{align*} \sum_{k=m_{1}}^{m_{2}}\cosh\left(ak+b\right) & =\sum_{k=m_{1}}^{m_{2}}\cosh\left(i\left(ak+b\right)\right)\\ & =\sum_{k=m_{1}}^{m_{2}}\cosh\left(iak+ib\right)\\ & =\sin^{-1}\left(\frac{ia}{2}\right)\cos\left(\left(m_{1}+m_{2}\right)\frac{ia}{2}+ib\right)\sin\left(\left(1+m_{2}-m_{1}\right)\frac{ia}{2}\right)\cmt{\because\sum_{k=m_{1}}^{m_{2}}\cos\left(ak+b\right)=\sin^{-1}\left(\frac{a}{2}\right)\cos\left(\left(m_{1}+m_{2}\right)\frac{a}{2}+b\right)\sin\left(\left(1+m_{2}-m_{1}\right)\frac{a}{2}\right)}\\ & =i^{-1}\sinh^{-1}\left(\frac{a}{2}\right)\cosh\left(\left(m_{1}+m_{2}\right)\frac{a}{2}+b\right)i\sinh\left(\left(1+m_{2}-m_{1}\right)\frac{a}{2}\right)\\ & =\sinh^{-1}\left(\frac{a}{2}\right)\cosh\left(\left(m_{1}+m_{2}\right)\frac{a}{2}+b\right)\sinh\left(\left(1+m_{2}-m_{1}\right)\frac{a}{2}\right) \end{align*}
ページ情報
| タイトル | 3角関数・双曲線関数の総和 |
| URL | https://www.nomuramath.com/xnco8o0d/ |
| SNSボタン |
巾関数と逆三角関数・逆双曲線関数の積の積分
\[
\int z^{\alpha}\Sin^{\bullet}zdz=\frac{1}{\alpha+1}\left(z^{\alpha+1}\Sin^{\bullet}z-\frac{z^{\alpha+2}}{\alpha+2}F\left(\frac{1}{2},\frac{\alpha}{2}+1;\frac{\alpha}{2}+2;z^{2}\right)\right)+C
\]
三角関数と双曲線関数の対数
\[
\log\sin x=-\log2+\frac{\pi}{2}i-ix-Li_{1}\left(e^{2ix}\right)
\]
三角関数の部分分数展開
\[
\pi\tan\pi x =-\sum_{k=-\infty}^{\infty}\frac{1}{x+\frac{1}{2}+k}
\]
三角関数と双曲線関数の実部と虚部
\[
\sin z=\sin\left(\Re\left(z\right)\right)\cosh\left(\Im\left(z\right)\right)+i\cos\left(\Re\left(z\right)\right)\sinh\left(\Im\left(z\right)\right)
\]