Was du bei fast jeder Aufgabe brauchst: Winkelwerte, trigonometrische Abkürzungen, Wurzeln, Arcus-Funktionen. Lieber hier nachschauen als raten.
Merkreihe für den Sinus: \(\tfrac{\sqrt0}{2},\tfrac{\sqrt1}{2},\tfrac{\sqrt2}{2},\tfrac{\sqrt3}{2},\tfrac{\sqrt4}{2}\). Der Cosinus läuft sie rückwärts.
| \(x\) | \(0\) | \(\tfrac\pi6\) | \(\tfrac\pi4\) | \(\tfrac\pi3\) | \(\tfrac\pi2\) | \(\pi\) |
|---|---|---|---|---|---|---|
| \(\sin x\) | \(0\) | \(\tfrac12\) | \(\tfrac{\sqrt2}{2}\) | \(\tfrac{\sqrt3}{2}\) | \(1\) | \(0\) |
| \(\cos x\) | \(1\) | \(\tfrac{\sqrt3}{2}\) | \(\tfrac{\sqrt2}{2}\) | \(\tfrac12\) | \(0\) | \(-1\) |
| \(\tan x\) | \(0\) | \(\tfrac{1}{\sqrt3}\) | \(1\) | \(\sqrt3\) | nicht definiert | \(0\) |
Für ganzzahliges \(n\): \(\cos(n\pi)=(-1)^n\) und \(\sin(n\pi)=0\).
Die Logarithmus-Regeln gelten für \(a,b>0\) und \(x>0\).
| \(f(x)\) | \(f'(x)\) | \(f(x)\) | \(f'(x)\) |
|---|---|---|---|
| \(x^n\) | \(n\,x^{n-1}\) | \(\sqrt x\) | \(\dfrac{1}{2\sqrt x}\) |
| \(\ee^x\) | \(\ee^x\) | \(\ln x\) | \(\dfrac1x\) |
| \(\sin x\) | \(\cos x\) | \(\cos x\) | \(-\sin x\) |
| \(\tan x\) | \(\dfrac{1}{\cos^2x}\) | \(\arctan x\) | \(\dfrac{1}{1+x^2}\) |
| \(\arcsin x\) | \(\dfrac{1}{\sqrt{1-x^2}}\) | \(\arccos x\) | \(-\dfrac{1}{\sqrt{1-x^2}}\) |
| \(f(x)\) | \(\int f\,\dd x\) | \(f(x)\) | \(\int f\,\dd x\) |
|---|---|---|---|
| \(x^n\ (n\neq-1)\) | \(\dfrac{x^{n+1}}{n+1}\) | \(\dfrac1x\) | \(\ln|x|\) |
| \(\ee^{ax}\) | \(\dfrac{\ee^{ax}}{a}\) | \(\ln x\) | \(x\ln x-x\) |
| \(\sin(ax)\) | \(-\dfrac{\cos(ax)}{a}\) | \(\cos(ax)\) | \(\dfrac{\sin(ax)}{a}\) |
| \(\tan x\) | \(-\ln|\cos x|\) | \(\dfrac{1}{\cos^2x}\) | \(\tan x\) |
| \(\dfrac{1}{1+x^2}\) | \(\arctan x\) | \(\dfrac{1}{a^2+x^2}\) | \(\dfrac1a\arctan\dfrac xa\) |
| \(\dfrac{1}{\sqrt{1-x^2}}\) | \(\arcsin x\) | \(\dfrac{1}{\sqrt{a^2-x^2}}\ (a>0)\) | \(\arcsin\dfrac xa\) |
| \(\sin^2x\) | \(\dfrac x2-\dfrac{\sin(2x)}{4}\) | \(\cos^2x\) | \(\dfrac x2+\dfrac{\sin(2x)}{4}\) |
Bei unbestimmten Integralen kommt jeweils \(+C\) dazu. In den Zeilen mit \(a\) ist \(a\neq0\).