数学の基礎と公式-2
数学の基礎と公式
amazon kindle版を出版しました。
2. 三角関数
(1)$\sin^2 x +\cos ^2 x=1$
(2)$\cot x=\dfrac{1}{\tan x} $
(3)$\sin x=\dfrac{1}{cosec x} $
※mathjaxにcosecはありません。
(4)$\cos x=\dfrac{1}{\sec x} $
(5)$\sec^2 x-\tan^2 x=1$
(6)$cosec^2 x-\cot^2 x=1$
(7)$\sin x=\cos x \tan x=2 \sin \cfrac{x}{2} \cos \cfrac{x}{2} \\ =\cot \dfrac{x}{2} (1-\cos x) =\tan \dfrac{x}{2} (1+\cos x)$
(8)$\cos x=\cos^2 \cfrac{x}{2}-\sin^2 \cfrac{x}{2}=2 \cos^2 \cfrac{x}{2}-1=\sin x \cot x$
(9)$\tan x=\dfrac{\sin 2x}{1+\cos 2x }=\dfrac{1-\cos 2x}{\sin 2x }=\dfrac{2 \tan \dfrac{x}{2}}{1-\tan^2 \dfrac{x}{2} }=\dfrac{\sec x}{cosec x}$
加法定理
(1)$\sin (x \pm y)=\sin x \cos y \pm \cos x \sin y$
(2)$\cos (x \pm y)=\cos x \cos y \mp \sin x \sin y$
(3)$\tan(x \pm y)=\dfrac{\tan x \pm \tan y}{1 \mp \tan x \tan y }$
(4)$\sin x \pm \sin y=2 \sin \dfrac{x \pm y}{2} \cos \dfrac{x \mp y}{2}$
(5)$\cos x + \cos y=2 \cos \dfrac{x+y}{2} \cos \dfrac{x - y}{2}$
(6)$\cos x - \cos y=2 \sin \dfrac{x+y}{2} \sin \dfrac{-x+ y}{2}$
(7)$2\sin x \cos y=\sin(x+y) +\sin(x-y)$
(8)$2\cos x \cos y=\cos(x+y) +\cos(x-y)$
(9)$2\sin x \sin y=-\cos(x+y) +\cos(x+y)$
(10)$\sin(x+y+z) \\=\sin x \cos y \cos z+\cos x \sin y \cos z+\cos x \cos y \sin z-\sin x \sin y \sin z$
(11)$\cos(x+y+z) \\=\cos x \cos y \cos z-\sin x \sin y \cos z-\sin x \cos y \sin z-\cos x \sin y \sin z$
(12)$\tan(x+y+z)=\dfrac{(- \tan z \tan y \tan x ) + \tan x + \tan y + \tan z }{(- \tan y \tan x ) - \tan z \tan x - \tan z \tan y + 1 }$
(13)$4 \sin x \sin y \cos z\\=- \cos(x + y + z) - \cos(x + y - z) + \cos(x - y + z) + \cos(x - y - z)$
(14)$4 \sin x \cos y \cos z\\= \sin(x + y + z) + \sin(x + y - z) + \sin(x - y + z) + \sin(x - y - z) $
(15)$4 \cos x \cos y \cos z\\=\cos(x + y + z) + \cos(x + y - z) + \cos(x - y + z) + \cos(x - y - z) $
倍角の公式
(1)$\sin 2x =2 \cos x \sin x $
(2)$\cos 2x=\cos^2 x - \sin^2 x$
(3)$\tan 2x=\dfrac{2 \tan x}{1-\tan^2 x}$
(4)$\sin 3x=3 \cos^2 x \sin x - \sin^3 x $
(5)$\cos 3x=\cos^3 x - 3 \cos x \sin^2 x $
(6)$\tan 3x=\dfrac{ 3 \tan x - \tan^3 x}{1 - 3 \tan^2 x} $
(7)$\sin 4x=4 \cos^3 x \sin x - 4 \cos x \sin^3 x $
(8) $\cos 4x=\sin^4 x - 6 \cos^2 x \sin^2 x + \cos^4 x $
(9) $\tan 4x=\dfrac{ 4 \tan x -4 \tan^3 x}{\tan^4 x - 6 \tan^2 x + 1 } $
(10) $\sin 5x=\sin^5 x - 10 \cos^2 x \sin^3 x + 5 \cos^4 x \sin x $
(11) $\cos 5x=5 \cos x \sin ^4 x-10 \cos ^3 x \sin ^2x+\cos ^5 x$
(12) $\tan 5 x={\dfrac{\tan ^5 x-10 \tan ^3 x+5 \tan x}{5 \tan ^4 x-10 \tan ^2x+1
}}$
(13) $\sin 6 x=6 \cos x \sin ^5 x-20 \cos ^3 x \sin ^3 x+6 \cos ^5 x \sin x$
(14) $\cos 6 x=-\sin ^6 x+15 \cos ^2 x \sin ^4 x-15 \cos ^4 x \sin ^2 x+\cos ^6 x$
(15) $\tan 6 x={\dfrac{6 \tan ^5 x-20 \tan ^3 x+6 \tan x}{-\tan ^6 x+15 \tan ^4 x- 15 \tan ^2 x+1}}$
(16) $\sin 7 x \\=-\sin ^7 x+21 \cos ^2 x \sin ^5 x-35 \cos ^4 x \sin ^3 x+7 \cos ^6 x \sin x $
(17) $\cos 7 x \\=-7 \cos x \sin ^6x+35 \cos ^3x \sin ^4x-21 \cos ^5x \sin ^2x+ \cos ^7x$
(18) $\tan 7 x={\dfrac{-\tan ^7 x+21 \tan ^5 x-35 \tan ^3 x+7 \tan x}{-7 \tan ^6 x+ 35 \tan ^4 x-21 \tan ^2 x+1}}$
(19) $\sin 8 x \\=-8 \cos x \sin ^7x+56 \cos ^3x \sin ^5x-56 \cos ^5x \sin ^3x+ 8 \cos ^7x \sin x$
(20) $\cos 8 x \\=\sin ^8x-28 \cos ^2x \sin ^6x+70 \cos ^4x \sin ^4x-28 \cos ^6x \sin ^2x+\cos ^8x$
(21) $\tan 8 x ={\dfrac{-8 \tan ^7x+56 \tan ^5x-56 \tan ^3x+8 \tan x}{\tan ^8x- 28 \tan ^6x+70 \tan ^4x-28 \tan ^2x+1}}$
(22) $\sin 9 x \\=\sin ^9x-36 \cos ^2x \sin ^7x+126 \cos ^4x \sin ^5x-84 \cos ^6 x \sin ^3x+9 \cos ^8x \sin x$
(23) $\cos 9 x \\=9 \cos x \sin ^8x-84 \cos ^3x \sin ^6x+126 \cos ^5x \sin ^4x- 36 \cos ^7x \sin ^2x+\cos ^9x$
(24) $\tan 9 x ={\dfrac{\tan ^9x-36 \tan ^7x+126 \tan ^5x-84 \tan ^3x+9 \tan x}{ 9 \tan ^8x-84 \tan ^6x+126 \tan ^4x-36 \tan ^2x+1}}$
(25) $\sin 10 x \\=10 \cos x \sin ^9x-120 \cos ^3x \sin ^7x+252 \cos ^5x \sin ^5 x-120 \cos ^7x \sin ^3x+10 \cos ^9x \sin x$
(26) $\cos 10 x \\=-\sin ^{10}x+45 \cos ^2x \sin ^8x-210 \cos ^4x \sin ^6x+210 \cos ^6x \sin ^4x-45 \cos ^8x \sin ^2x+\cos ^{10}x$
(27) $\tan 10 x ={\dfrac{10 \tan ^9x-120 \tan ^7x+252 \tan ^5x-120 \tan ^3x+10 \tan x }{-\tan ^{10}x+45 \tan ^8x-210 \tan ^6x+210 \tan ^4x-45 \tan ^2x+1}}$
