From a3b1fbefd8707113d9c350ab7dc053be19856c92 Mon Sep 17 00:00:00 2001 From: zacharias Date: Tue, 11 Nov 2025 17:01:13 +0100 Subject: [PATCH] vault backup: 2025-11-11 17:01:13 --- .obsidian/workspace.json | 23 ++++++++++++++++++++--- Komplexa tal.md | 24 ++++++++++++++++++++++++ Trigonometri.md | 11 ++++++++++- k1.png | Bin 0 -> 54988 bytes k2.png | Bin 0 -> 54215 bytes 5 files changed, 54 insertions(+), 4 deletions(-) create mode 100644 Komplexa tal.md create mode 100644 k1.png create mode 100644 k2.png diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json index f9318fb..26ee7d9 100644 --- a/.obsidian/workspace.json +++ b/.obsidian/workspace.json @@ -62,9 +62,23 @@ "icon": "lucide-file", "title": "Trigonometri" } + }, + { + "id": "be47d5ede3a9176b", + "type": "leaf", + "state": { + "type": "markdown", + "state": { + "file": "Komplexa tal.md", + "mode": "source", + "source": false + }, + "icon": "lucide-file", + "title": "Komplexa tal" + } } ], - "currentTab": 3 + "currentTab": 4 } ], "direction": "vertical" @@ -224,10 +238,13 @@ "obsidian-git:Open Git source control": false } }, - "active": "4eef5f8feb086f9e", + "active": "be47d5ede3a9176b", "lastOpenFiles": [ - "Funktioner Forts.md", + "k2.png", + "k1.png", "Trigonometri.md", + "Komplexa tal.md", + "Funktioner Forts.md", "Grafer.md", "Funktioner.md", "f_inverse.png", diff --git a/Komplexa tal.md b/Komplexa tal.md new file mode 100644 index 0000000..54f8602 --- /dev/null +++ b/Komplexa tal.md @@ -0,0 +1,24 @@ +- Komplexa tal + - **Def**: $x^2+1=0$ saknar reell lösning. Vi antar talet $i\notin\mathbb{R}$ löser ekvationen, d.v.s $i^2=-1$ + - Mängd av komplexa talen: $\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}$ + - Om $z=a+bi,a=Re(z)$ och $b=Im(z)$ + - **Konjugat**: $z=a+bi\Rightarrow\bar{z}=a-bi$ + - **Regler**: + - $\bar{\bar{z}}=z$ + - $\overline{z_1+z_2}=\overline{z_1}+\overline{z_2}$ + - $\overline{z_1\times{z_2}}=\overline{z_1}\times{z_2}$ + - **Absolut belopp**: $$\mid{z}\mid=\mid\overline{z}\mid=\sqrt{z\overline{z}}=\sqrt{a^2+b^2}\text{ om }z=a+bi$$ + - **Triangelsformeln**: $\mid{z_1+z_2}\mid\leq\mid{z_1}\mid+\mid{z_2}\mid$ + - **Ex**: $$\begin{align}z_1=2+3i\\z_2=2-i\\\\z_1+z_2=(2+3i)+(2-1)\\=4+2i\\\overline{z_1+z_2}=4-2i\\\overline{z_1}=2-3i,\;\overline{z_2}=2+i\\\overline{z_1}+\overline{z_2}=2-3i+2+i\\=3-2i\\\\z_1\times{z_2}=(2+3i)(2-i)\\=4-2i+6i-3i^2\\=4+4i+3\\=7+4i\\\overline{z_1\times{z_2}}=7-4i\\\overline{z_1}=2-3i,\;\overline{z_2}=2+i\\\overline{z_1}\times\overline{z_2}=(2-3i(2+i)\\=4+2i-6i-3i^2\\=4-2i+3\\=7-4i\end{align}$$ + - **Ex 2**: $$\begin{align}z=a+bi\\\overline{z}=a-bi\\z\times\overline{z}=(a+bi)(a-bi)\\=a^2-\left(bi\right)^2\\=a^2-b^2i^2\\=a^2+b^2\end{align}$$ + - **Ex 3**: $$\begin{align}\mid{z_1+z_2}\mid=\mid4+2i\mid\\=\sqrt{4^2+2^2}\\=\sqrt{16+4}=2\sqrt{5}\\\mid{z_1}\mid=\mid2+3i=\sqrt{2^2+3^2}\\=\sqrt{13}\\\mid{z_2}\mid=\mid2-i\mid=\sqrt{2^2+(-i)^2}=\sqrt{5}\end{align}$$ + - **Ex 4**: $$\begin{align}\frac{z_1}{z_2}=\frac{2+3i}{2-i}\\=\frac{2+3i}{2-i}\times\frac{2+i}{2+i}\\=\frac{4+2i+6i+3i^2}{2^2-i^2}\\=\frac{1+8i}{5}\\=\frac{1}{5}+\frac{8}{5}i\end{align}$$ +- Grafer + - ![[k1.png]] + - ![[k2.png]] +- Polär form + - **Eulers formel**: $e^{i\theta}=\cos\theta+i\sin\theta$ + - Varje komplex tal $z=x+yi$ kan skrivas på pol'r form som $$z=re^{i\theta}$$ där $$r=\sqrt{x^2+y^2}$$ och $arg(z)=\theta$ är så att $$\cos\theta=\frac{x}{\sqrt{x^2+y^2}}\text{ och }\sin\theta=\frac{y}{x^2+y^2}$$ + - **de Moivre**: $z=re^{i\theta}\Rightarrow z^n=r^ne^{in\theta}=r^n(\cos(n\theta)+i\sin(n\theta))$ + - **Ex**: Lös $z^3=1+i\sqrt3$ $$\begin{align}r=\sqrt{1^2+\sqrt3^2}\\=\sqrt{1+3}\\=\sqrt{4}\\=2\\\\\end{align}$$ + - **Ex 2**: $$\begin{align}z=-\frac{\sqrt3}{2}+\frac{1}{2}i\\z=ne^{i\theta}\\n=\sqrt{\left(-\frac{\sqrt{3}}{2}\right)^2+\left(\frac{1}{2}\right)^2}=1\\\theta\text{ är så att }\cos\theta=\frac{\sqrt{3}}{2},\sin\theta=\frac{1}{2}\\\text{En lösning}:\end{align}$$ \ No newline at end of file diff --git a/Trigonometri.md b/Trigonometri.md index 67d393a..44b1dac 100644 --- a/Trigonometri.md +++ b/Trigonometri.md @@ -22,4 +22,13 @@ - $\sin\theta=\sin{a}\Leftrightarrow\theta=\left\{\begin{align}a+2n\pi,n\in\mathbb{Z}\\\pi-a+2n\pi,n\in\mathbb{Z}\end{align}\right.$ - $\cos\theta=\cos{a}\Leftrightarrow\theta=\left\{\begin{align}a+2n\pi,n\in\mathbb{Z}\\-a+2n\pi,n\in{Z}\end{align}\right.$ - $\tan\theta=\tan{a}\Leftrightarrow\theta=a+n\pi,n\in\mathbb{Z}$ - - Ex: Solve $\sin(x+\frac{\pi}{6})=\frac{\sqrt{3}}{2}$ \ No newline at end of file + - 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. $$\begin{align}\arcsin(-x)=-\arcsin(x)\\\arccos(-x)=\pi-\arccos(x)\\\arctan(-x)=-\arctan(x)\end{align}$$* +- \ No newline at end of file diff --git a/k1.png b/k1.png new file mode 100644 index 0000000000000000000000000000000000000000..09d92a1b85a915fb2034611ba14b8d739e7750f3 GIT binary patch literal 54988 zcmeAS@N?(olHy`uVBq!ia0y~yVA5k?V7$)3#=yWZ+y3HY1_lPs0*}aI1_r((Aj~*b zn@^g7fq|#QHKHUZKRq)!F(-n-$jsb-U-@D*@kjfduK5rg6(vD!=&Gu5&!KMJQ1{d{hxi#t=kMLI;Wn+_ZX%` zmMQMgdDLos+UQ6jPvo?PuYVo-`_F0CvW~|Rt5)gVv1DA3-CXc*XMo;x{ii=aoH?j^ z+Q=oa#iCEqQz+-Q+upeSQ{=N|wI*`h|K2@$Ywqn0I@yonKCi$0Qh~>LikIq@*I!Ta zpEuR)PGE_hanF6jV`q<5jmzW?uJSmsRce9GF;DB$MnCrTTl6t9>{@qt+uPRat+H*3 zZHE<8BHdPoG_89b5L%jet~YX;qNfmxV}s6Vu}^at7#sx-e6MA&IdXd&?`0G9cc%9x 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