三角関数

倍角の公式の導出メモ

$ sin(2a) = sin(a)cos(a)+cos(a)sin(a) = 2sin(a)cos(a)

$ cos(2a) = cos(a)cos(a)-sin(a)sin(a) = cos^2(a)-sin^2(a) = cos^2(a) - (1 - cos^2(a)) = 2cos^2(a) -1

$ cos(2a) = cos^2(a)-sin^2(a) = (1 - sin^2(a))-sin^2(a) = 1 - 2sin^2(a)

t=tanx/2の置換

$ t = tan\frac{x}{2}

倍角の公式と半角の公式を使う。

$ sinx = \frac{2t}{1+t^2}

$ cosx = \frac{1-t^2}{1+t^2}

$ dx = \frac{2}{1+t^2} dt

半角（2乗）の公式

$ cos^2(a) = \frac{1+cos(2a)}{2}

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

2乗の微分

$ (sin^2x)' = sin2x

$ (cos^2x)' = -2sinxcosx

$ (tan2x)' = \frac{2sinx}{cos^3x}

分数

$ \int \frac{1}{sinx} = \log|tan\frac{x}{2}| + C

逆関数

$ (sin^{-1}x)' = \frac{1}{\sqrt{1-x^2}}

$ (cos^{-1}x)' = - \frac{1}{\sqrt{1-x^2}}

$ (tan^{-1}x)' = \frac{1}{1+x^2}