삼각함수

1 덧셈 정리

$$\begin{array}{r}\sin (A+B)=\sin A\cos B+\cos A\sin B\\ \cos (A+B)=\cos A\cos B-\sin A\sin B\end{array}$$

2 미분

2.1 삼각함수 미분

2.1.1 sin

$$\begin{array}{rl}f(x)& =\sin x\\ f^{\prime }(x)& =\underset{h\to 0}{\lim }\frac{\sin (x+h)-\sin x}{h}\\ & =\underset{h\to 0}{\lim }\frac{\sin x\cos h+\cos x\sin h-\sin x}{h}\\ & =\underset{h\to 0}{\lim }\frac{\sin x(\cos h-1)+\cos x\sin h}{h}\\ & =\sin x\underset{h\to 0}{\lim }\frac{\cos h}{h}+\cos x\underset{h\to 0}{\lim }\frac{\sin h}{h}\\ & =\cos x\end{array}$$

2.1.2 cos

$$\begin{array}{rl}f(x)& =\cos x\\ f^{\prime }(x)& =\underset{h\to 0}{\lim }\frac{\cos (x+h)-\cos x}{h}\\ & =\underset{h\to 0}{\lim }\frac{\cos x\cos h-\sin x\sin h-\cos x}{h}\\ & =\underset{h\to 0}{\lim }\frac{\cos x(\cos h-1)-\sin x\sin h}{h}\\ & =\cos x\underset{h\to 0}{\lim }\frac{\cos h-1}{h}-\sin x\underset{h\to 0}{\lim }\frac{\sin h}{h}\\ & =-\sin x\end{array}$$

2.1.3 tan

$$\begin{array}{rl}f(x)& =\tan x\\ f^{\prime }(x)& ={\left(\frac{\sin x}{\cos x}\right)}^{\prime }\\ & =\frac{(\sin x{)}^{\prime }\cos x+(\cos x{)}^{\prime }\sin x}{{\cos }^{2}x}\\ & =\frac{1}{{\cos }^{2}x}\\ & =1+{\tan }^{2}x\\ & ={\sec }^{2}x\end{array}$$

2.2 역삼각함수 미분

역함수 미분

2.2.1 arcsin

$$\frac{1}{\sqrt{1-x^2}}$$

2.2.2 arccos

$-\frac{1}{\sqrt{1-x^2}}$

2.2.3 arctan

$\frac{1}{1+x^2}$

3 ${\int }_a^b\sin xdx={\int }_a^b\cos xdx$

정적분의 대칭성을 통해 유도할 수 있다.

4 Unknown1

$$\begin{array}{rl}& \\ & {\sin }^{n}x\le {\sin }^{n-1}x\\ & {\int }_0^{\pi /2}{\sin }^{n}xdx\le {\int }_0^{\pi /2}{\sin }^{n-1}xdx\\ & where0<x<\pi \end{array}$$