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Merge branch 'master' of git.server.4zellen.se:Zacharias/Analys-och-Linj-r-algibra

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Zacharias Zellén
2025-11-12 17:01:33 +01:00
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Trigonometri.md
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- $\sin\theta=\sin{a}\Leftrightarrow\theta=\left\{\begin{align}a+2n\pi,n\in\mathbb{Z}\\\pi-a+2n\pi,n\in\mathbb{Z}\end{align}\right.$
- $\cos\theta=\cos{a}\Leftrightarrow\theta=\left\{\begin{align}a+2n\pi,n\in\mathbb{Z}\\-a+2n\pi,n\in{Z}\end{align}\right.$
- $\tan\theta=\tan{a}\Leftrightarrow\theta=a+n\pi,n\in\mathbb{Z}$
- Ex: Solve $\sin(x+\frac{\pi}{6})=\frac{\sqrt{3}}{2}$
- Ex: Solve $\sin(x+\frac{\pi}{6})=\frac{\sqrt{3}}{2}$
- Inverse trigonometric function
- **Def**: *$f(x)=\sin(x),x\in\left[\frac{-\pi}{2},\frac{\pi}{2}\right]$. Then $f$ is strictly increasing on $D_f$ and hence inverible. The fuction $\arcsin$ is defined as $$\arcsin(x)=f^{-1}(x)\text{ on }D_{arcsin}=R_f=[-1,1]$$*
- **Similarly**: *For $g(x)=\cos(x),x\in\left[0,\pi\right]$ (which is strictly decreasing) and $h(x)=\tan(x),x\in\left[\frac{-\pi}{2},\frac{\pi}{2}\right]$ (which is strictly increasing), the function $\arccos$ and $\arctan$ are defined as $$\begin{align}\arccos(x)=g^{-1}(x)\text{ on }D_{\arccos}=\left[-1,1\right]\\\arctan(x)=h^{-1}(x)\text{ on }D_{\arctan}=\mathbb{R}\end{align}$$*
- **Note**: *That the tanges $R_{\arcsin}=R_{\arctan}=\left[\frac{-\pi}{2},\frac{\pi}{2}\right]$ whereas $R_{\arccos}=\left[0,\pi\right]$*
- Properties
- **Def**: $$\begin{align}\sin(\arcsin(x))=x\forall{x}\in\left[-1,1\right]\text{ | }\arcsin(sin(x))=x\text{ if }x\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\\\cos(\arccos(x))=x\forall{x}\in\left[-1,1\right]\text{ | }\arccos(\cos(x))=x\text{ if }x\in\left[0,\pi\right]\\\tan(\arctan(x))=x\forall{x}\in\mathbb{R}\text{ | }\arctan(\tan(x))=x\text{ if }x\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\end{align}$$
- **Complementary angles**: $$\arcsin(x)+\arccos(x)=\frac{pi}{2},\;\arctan(x)+\arccos(x)=\frac{\pi}{2}$$
- **Negatives**: *$\arcsin$ and $\arctan$ are odd functions. $$\begin{align}\arcsin(-x)=-\arcsin(x)\\\arccos(-x)=\pi-\arccos(x)\\\arctan(-x)=-\arctan(x)\end{align}$$*
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