Trigonometriska formler
Definitioner
\[ \sin(x) = \frac{1}{\sec(x)} \]
\[ \cos(x) = \frac{1}{\csc(x)} \]
\[ \tan(x) = \frac{1}{\cot(x)} \]
\[ \tan(x) = \frac{\sin(2x)}{1 + \cos(2x)} = \frac{1-\cos(2x)}{\sin(2x)} \]
\[ \cot(x) = \frac{\sin(2x) }{1 - \cos(2x)} = \frac{1 + \cos(2x)}{\sin(2x)} \]
Enhetscirkeln på olika sätt
\[\sin^2(x) + \cos^2(x) = 1 \] \[1 + \tan^2(x) = \sec^2(x)\] \[ 1 + \cot^2(x) = \csc^2(x) \]
Enhetscirkeln på olika sätt
\[\sin^2(x) + \cos^2(x) = 1 \] \[1 + \tan^2(x) = \sec^2(x)\] \[ 1 + \cot^2(x) = \csc^2(x) \]
Additionsformlerna
Sin
\[ \sin(x + y) = \sin(x)\cos(y) + \sin(y)\cos(x) \]
\[ \sin(x - y) = \sin(x)\cos(y) - \sin(y)\cos(x) \]
Cos
\[ \cos(x + y) = \cos(x)\cos(y) - \sin(x)\sin(y) \]
\[ \cos(x - y) = \cos(x)\cos(y) + \sin(x)\sin(y) \]
Tan
\[ \tan(x + y) = \frac{\tan(x) + \tan(y) }{ 1 - \tan(x) \tan(y)} \]
\[ \tan(x - y) = \frac{\tan(x) - \tan(y) }{1 + \tan(x) \tan(y)} \]
Dubbla vinkeln
\[ \sin(2x) = 2\sin(x)\cos(x) \]
\[ \cos(2x) = \cos^2(x) - \sin^2(x) = 2\cos^2(x) -1 = 1- 2\sin^2(x)\]
\[ \tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)} \]
\[ \cot(2x) = \frac{\cot(x) - \tan(x)}{2} \]
Halva vinkeln
\[ \sin^2\left( \frac{x}{2} \right) = \frac{1 - \cos(x)}{2} \]
\[\cos^2\left( \frac{x}{2} \right) = \frac{1 + \cos(x)}{2} \]
\[ \tan^2\left( \frac{x}{2} \right) = \frac{1 - \sin(x)}{1+\cos(x)} \]
\[ \cot^2\left( \frac{x}{2}\right) = \frac{1 + \cos(x)}{1-\cos(x)}\]
Adddition och subtraktion
\[ \sin(x) + \sin(y) = 2\sin\left( \frac{x + y}{2} \right)\cos\left(\frac{x - y }{2} \right)\]
\[ \sin(x) - \sin(y) = 2\sin\left( \frac{x - y}{2} \right)\cos\left(\frac{x + y }{2} \right)\]
\[ \cos(x) + \cos(y) = 2\cos\left( \frac{x + y}{2} \right)\cos\left(\frac{x - y }{2} \right)\]
\[ \cos(x) - \cos(y) = -2\sin\left( \frac{x + y}{2}\right)\sin\left(\frac{x - y }{2} \right)\]
Translation och perioder
\[ \sin(-x) = -\sin(x) \]
\[ \cos(-x) = \cos(x) \]
\[ \tan(-x) = -\tan(x) \]
\[ \cot(-x) = - \cot(x) \]
\[ \sec(-x) = \sec(x) \]
\[ \csc(-x) = -\csc(x) \]
\[ \sin(x + 2\pi n) = \sin(x) \]
\[ \cos(x + 2\pi n) = \cos(x) \]
\[ \tan(x + \pi n) = \tan(x) \]
\[ \cot(x + \pi n) = \cot(x) \]
\[ \sec(x + 2\pi n) = \sec(x) \]
\[ \csc(x + 2\pi n) = \csc(x) \]
\[ \sin\left( \frac{\pi}{2} - x\right) = \cos(x) \]
\[ \cos\left(\frac{\pi}{2} - x\right) = \sin(x) \]
\[ \tan\left( \frac{\pi}{2} - x\right) = \cot(x) \]
\[ \cot\left(\frac{\pi}{2} - x\right) = \tan(x) \]
\[ \sec\left( \frac{\pi}{2} - x\right) = \csc(x) \]
\[ \csc\left(\frac{\pi}{2} - x\right) = \sec(x) \]
\[ \sin(\pi - x) = \sin(x) \]
\[ \cos(\pi - x) = -\cos(x)\]
\[ \tan(\pi - x) = -\tan(x) \]
\[ \cot(\pi - x) = -\cot(x) \]
\[ \sec(\pi - x) = - \sec(x) \]
\[ \csc(\pi - x) = \csc(x) \]
\[ \cos(x) = \sin\left( x + \frac{\pi}{2} \right) \]
\[ \tan(x) = \frac{\sin(x)}{\cos(x)} = \tan\left( \frac{\pi}{2} - x \right) \]
\[ \sec(x) = \frac{1}{\cos(x)} \]
\[ \csc(x) = \frac{1}{\sin(x)} \]