vault backup: 2025-11-04 16:53:01

.obsidian/workspace.json

@@ -34,9 +34,37 @@
               "icon": "lucide-file",
               "title": "Funktioner Forts"
             }
+          },
+          {
+            "id": "54baa3edd65a7c5d",
+            "type": "leaf",
+            "state": {
+              "type": "markdown",
+              "state": {
+                "file": "Grafer.md",
+                "mode": "source",
+                "source": false
+              },
+              "icon": "lucide-file",
+              "title": "Grafer"
+            }
+          },
+          {
+            "id": "4eef5f8feb086f9e",
+            "type": "leaf",
+            "state": {
+              "type": "markdown",
+              "state": {
+                "file": "Trigonometri.md",
+                "mode": "source",
+                "source": false
+              },
+              "icon": "lucide-file",
+              "title": "Trigonometri"
+            }
           }
         ],
-        "currentTab": 1
+        "currentTab": 3
       }
     ],
     "direction": "vertical"
@@ -93,7 +121,8 @@
       }
     ],
     "direction": "horizontal",
-    "width": 300
+    "width": 300,
+    "collapsed": true
   },
   "right": {
     "id": "b700e0cd0f882a5c",
@@ -195,14 +224,15 @@
       "obsidian-git:Open Git source control": false
     }
   },
-  "active": "66704e0159322e3f",
+  "active": "4eef5f8feb086f9e",
   "lastOpenFiles": [
-    "g2.png",
-    "Grafer.md",
-    "g1.png",
     "Funktioner Forts.md",
-    "f_inverse.png",
+    "Trigonometri.md",
+    "Grafer.md",
     "Funktioner.md",
+    "f_inverse.png",
+    "g2.png",
+    "g1.png",
     "Untitled.canvas"
   ]
 }

Funktioner Forts.md

@@ -36,3 +36,19 @@
 		- $f(x)=x^2,\;x\in[0,1]$ $D_f=[0,1]$
 		- ![[g2.png]]
 		- $$\begin{align}f(x)=3x+5\\g(x)=\frac{x-5}{3}\end{align}$$
+- Exponential och logarithm
+	- Exponential: $f(x)=a^x$ för något $a>0$.
+	- Logaritm: $g(x)=\log_a(x)$ för något $a>0$
+		- $f$ och $g$ inverse till varandra: $y=a^x\Leftrightarrow\log_a(y)=x$.
+		- $D_f=\mathbb{R}=V_g,\;\;V_f=(0,\infty)=D_g$.
+		- Om $a>1,\;f,\;g$ är strängt växande.
+		- $\log_a{(xy)}=\log_a(x)+\log_a(y),\;\log_a(x/y)=\log_a(x)-\log_a(y)$
+		- $\log_a(x^b)=b\log_a(x)$
+		- Basbyte: $\log_a(x)=\frac{\log_b(x)}{\log_b(a)}\Leftrightarrow\log_b(x)=\log_b(a)\log_a(x)$. $a^x=b^{x\log_b(a)}$
+		- Ex: $$\begin{align}\text{Räkna }D_f\text{ för }f(x)=\log_{10}(x^2+2x-3)\\f\text{ är definierad för }x^2+2x-3>0\\\Leftrightarrow(x+3)(x-1)>0\\\Leftrightarrow x\in(-\infty,-3)\cup(1,\infty)\\D_f=(-\infty,-3)\cup(1,\infty)\\\\2^{x+3}>4\\\Leftrightarrow\log_2(2^{x+3})>\log_24\\\Leftrightarrow x+3>2\\\Leftrightarrow x>-1\\\\\log_{10}36\\=\log_{10}(2^2\times3^2)\\=\log_{10}(2^2)+\log_{10}(3^2)\\=2\log_{10}2+2\log_{10}3\\\\2^x=e^{x\log_e2}=e^{x\ln2}\\\log_2x=(\log_2e)\ln{x}\\=\frac{\ln x}{\ln 2}\end{align}$$
+		- **Def**: $\log{x}=\log_{10}x$
+		- **Def**: $\ln{x}=\log_ex$
+		- **Def**: $a^x=e^{x\log_ea}=e^{x\ln a},\;a\in(0,\infty)$
+		- **Def**: $\log_a1=0$
+		- **Def**: $\log_aa=1$
+		- **Def**: $\log_ab=\frac{1}{\log_ba}$

Trigonometri.md

@@ -0,0 +1,25 @@
+- Radian:
+	- **Def**: *It is the SI unit for measuring angles (in the plane).*
+	- **Def**: *$1$ radian is defined as the angle subtended at the center by a circular arc of length equal to the radius*
+	- **Def**: *A general angle is measured in radians as the ration of the length an associated circular arc and the corresponding radius. That is $\theta=\frac{s}{r}\text{rad}$*
+	- **Def**: *Usually "$rad$" is omitted.*
+	- Ex: $$\begin{align}180^\circ=\pi\text{ rad}\\\frac{\pi}{3}\text{ rad}=30^\circ\\\frac{\pi}{4}\text{ rad}=45^\circ\\\frac{\pi}{3}\text{ rad}=60^\circ\\\frac{\pi}{2}\text{ rad}=90^\circ\\2\pi\text{ rad}=360^\circ\end{align}$$
+- The right angled triangle
+	- **Def**: *The trigonometric functions: *$$\begin{align}\sin\theta=\frac{\text{perpendicular}}{\text{hypotenuse}}\\\cos\theta=\frac{\text{base}}{\text{hypotenuse}}\\\tan\theta=\frac{\text{perpendicular}}{\text{base}}\end{align}$$
+	- Dominains and ranges:
+		- $D_{\sin}=\mathbb{R}\;\;R_{\sin}=[-1,1]$
+		- $D_{\cos}=\mathbb{R}\;\;R_{\cos}=[-1,1]$
+		- $D_{\tan}=\mathbb{R}\setminus\{n\pi+\frac{\pi}{2}:n\in\mathbb{Z}\}\;\;R_{\tan}=(-\infty,\infty)$
+- Useful relations
+	- $\sin(-\theta)=-\sin(\text{odd}),\cos(-\theta)=\cos\theta(\text{even})$
+	- Periodicity: $\sin(\theta+2n\pi)=\sin\theta,\cos(\theta+2n\pi)=\cos\theta,\tan(\theta+n\pi)=\tan\theta$
+	- Complementary angles: $\sin(\frac{\pi}{2}-\theta)=\cos\theta,\cos(\frac{\pi}{2}-\theta)=\sin\theta$
+	- Sift by $\pi$: $\sin(\theta\pm\pi)=-\sin\theta,\cos(\theta\pm\pi=-\cos\theta$
+	- Sum of angles: $\sin(\theta+\phi)=\sin\theta\times\cos\phi+\cos\theta\times\sin\phi,\cos(\theta+\phi)=\cos\theta\times\cos\phi-\sin\theta\times\sin\phi,\tan(\theta+\phi)=\frac{\tan\theta+\tan\phi}{1-\tan\theta\times\tan\phi}$
+	- Double angle: $\sin(2\theta)=2\sin\theta\cos\theta,\tan2\theta=\frac{2\tan\theta}{1-\tan^2\theta},\cos(2\theta)=\cos^2-\sin^2\theta=2\cos^2\theta-1=1-2\sin^2\theta$
+	- Half angle: $2\sin^2\frac{\theta}{2}=1-\cos\theta,2\cos^2\frac{\theta}{2}=1+\cos\theta$
+- Solving trigonometric equations
+	- $\sin\theta=\sin{a}\Leftrightarrow\theta=\left\{\begin{align}a+2n\pi,n\in\mathbb{Z}\\\pi-a+2n\pi,n\in\mathbb{Z}\end{align}\right.$
+	- $\cos\theta=\cos{a}\Leftrightarrow\theta=\left\{\begin{align}a+2n\pi,n\in\mathbb{Z}\\-a+2n\pi,n\in{Z}\end{align}\right.$
+	- $\tan\theta=\tan{a}\Leftrightarrow\theta=a+n\pi,n\in\mathbb{Z}$
+	- Ex: Solve $\sin(x+\frac{\pi}{6})=\frac{\sqrt{3}}{2}$