Trigonometriska formler

Definitioner

\[ \sin(x) = \frac{1}{\csc(x)} \]

\[ \cos(x) = \frac{1}{\sec(x)} \]

\[ \tan(x) = \frac{1}{\cot(x)} \]

 

\[ \tan(x) = \frac{\sin(2x)}{1 + \cos(2x)} = \frac{1-\cos(2x)}{\sin(2x)} \]

\[ \cot(x) = \frac{\sin(2x) }{1 - \cos(2x)} = \frac{1 + \cos(2x)}{\sin(2x)} \]

 

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

\[ \sin(-x) = -\sin(x) \]

\[ \cos(-x) = \cos(x) \]

\[ \tan(-x) = -\tan(x) \]

\[ \cot(-x) = - \cot(x) \]

\[ \sec(-x) = \sec(x) \]

\[ \csc(-x) = -\csc(x) \]

 

\[ \sin(x + 2\pi n) = \sin(x) \]

\[ \cos(x + 2\pi n) = \cos(x) \]

\[ \tan(x + \pi n) = \tan(x) \] 

\[ \cot(x + \pi n) = \cot(x) \]

\[ \sec(x + 2\pi n) = \sec(x) \]

\[ \csc(x + 2\pi n) = \csc(x) \]

 

\[ \sin\left( \frac{\pi}{2} - x\right) = \cos(x) \]

\[ \cos\left(\frac{\pi}{2} - x\right) = \sin(x) \]

\[ \tan\left( \frac{\pi}{2} - x\right) = \cot(x) \]

\[ \cot\left(\frac{\pi}{2} - x\right) = \tan(x) \]

\[ \sec\left( \frac{\pi}{2} - x\right) = \csc(x) \]

\[ \csc\left(\frac{\pi}{2} - x\right) = \sec(x) \]

\[ \sin(\pi - x) = \sin(x) \]

\[ \cos(\pi - x) = -\cos(x)\]

\[ \tan(\pi - x) = -\tan(x) \]

\[ \cot(\pi - x) = -\cot(x) \]

\[ \sec(\pi - x) = - \sec(x) \]

\[ \csc(\pi - x) = \csc(x) \]

 

\[ \cos(x) = \sin\left( x + \frac{\pi}{2} \right) \]

\[ \tan(x) = \frac{\sin(x)}{\cos(x)} = \tan\left( \frac{\pi}{2} - x \right) \]

\[ \sec(x) = \frac{1}{\cos(x)} \]

\[ \csc(x) = \frac{1}{\sin(x)} \]