Trigonometriske identiteter:

\(\Large sin(\alpha) = \dfrac{\text{motsatt katet}}{\text{hypotenus}} = \dfrac{tan(\alpha)}{sec(\alpha)} = \dfrac{1}{csc(\alpha)}\)

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\(\Large cos(\alpha) = \dfrac{\text{nærliggende katet}}{\text{hypotenus}} = \dfrac{cot(\alpha)}{csc(\alpha)} = \dfrac{1}{sec(\alpha)}\)

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\(\Large tan(\alpha) = \dfrac{\text{motsatt katet}}{\text{nærliggende katet}} = \dfrac{sin(\alpha)}{cos(\alpha)} = \dfrac{1}{cot(\alpha)}\)

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\(\Large sec(\alpha) = \dfrac{\text{hypotenus}}{\text{nærliggende katet}} = \dfrac{csc(\alpha)}{cot(\alpha)} = \dfrac{1}{cos(\alpha)}\)

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\(\Large cot(\alpha) = \dfrac{\text{nærliggende katet}}{\text{motsatt katet}} = \dfrac{cos(\alpha)}{sin(\alpha)} = \dfrac{1}{tan(\alpha)}\)

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\(\Large csc(\alpha) = \dfrac{\text{hypotenus}}{\text{motsatt katet}} = \dfrac{sec(\alpha)}{tan(\alpha)} = \dfrac{1}{sin(\alpha)}\)

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\(\Large sin^2 (\alpha) + cos^2 (\alpha) = 1\)

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\(\Large tan(\alpha) = \dfrac{sin(\alpha)}{cos(\alpha)}\)

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\(\Large tan(\alpha) \cdot cos(\alpha) = 1\)

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\(\Large sin(2\alpha) = 2 sin(\alpha) \cdot cos(\alpha)\)

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\(\Large cos(2\alpha) = cos^2(\alpha) - sin^2(\alpha) = 2 cos^2(\alpha) -1\)

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\(\Large cos(2\alpha) = 1 - 2 sin^2(\alpha)\)

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\(\Large tan(2\alpha) = \dfrac{2 tan(\alpha)}{1-tan^2(\alpha)}\)

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\(\Large sin(3 \alpha) = 3 sin(\alpha) - 4 sin^3(\alpha)\)

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\(\Large cos(3 \alpha) = 4 cos^3(\alpha) - 3 cos(\alpha)\)

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\(\Large sin(\alpha ± \beta) = sin(\alpha) \cdot cos(\beta) ± cos(\alpha) \cdot sin(\beta)\)

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\(\Large cos(\alpha + \beta) = cos(\alpha) \cdot cos(\beta) - sin(\alpha) \cdot sin(\beta)\)

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\(\Large cos(\alpha - \beta) = cos(\alpha) \cdot cos(\beta) + sin(\alpha) \cdot sin(\beta)\)

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\(\Large tan(\alpha + \beta) = \dfrac{tan(\alpha) + tan(\beta)}{1 - tan(\alpha) \cdot tan(\beta)}\)

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\(\Large tan(\alpha - \beta) = \dfrac{tan(\alpha) - tan(\beta)}{1 + tan(\alpha) \cdot tan(\beta)}\)

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\(\Large sin(\alpha) + sin(\beta) = 2 sin(\dfrac{\alpha + \beta}{2}) \cdot cos(\dfrac{\alpha + \beta}{2})\)

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\(\Large sin(\alpha) - sin(\beta) = 2 cos(\dfrac{\alpha + \beta}{2}) \cdot sin(\dfrac{\alpha + \beta}{2})\)

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\(\Large cos(\alpha) + cos(\beta) = 2 cos(\dfrac{\alpha + \beta}{2}) \cdot cos(\dfrac{\alpha + \beta}{2})\)

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\(\Large cos(\alpha) - cos(\beta) = -2 sin(\dfrac{\alpha + \beta}{2}) \cdot sin(\dfrac{\alpha + \beta}{2})\)

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\(\Large sin(\alpha) \cdot cos(\beta) = \dfrac{1}{2} \left( sin(\alpha + \beta) + sin(\alpha - \beta) \right)\)

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\(\Large sin(\beta) \cdot cos(\alpha) = \dfrac{1}{2} \left( sin(\alpha + \beta) - sin(\alpha - \beta) \right)\)

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\(\Large cos(\beta) \cdot cos(\alpha) = \dfrac{1}{2} \left( cos(\alpha + \beta) + cos(\alpha - \beta) \right)\)

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\(\Large sin(\beta) \cdot sin(\alpha) = \dfrac{1}{2} \left( cos(\alpha - \beta) - cos(\alpha + \beta) \right)\)

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For copy/paste

Markdown / Latex:

$\Large sin(\alpha) = \dfrac{\text{motsatt katet}}{\text{hypotenus}} = \dfrac{tan(\alpha)}{sec(\alpha)} = \dfrac{1}{csc(\alpha)}$

$\,$

$\Large cos(\alpha) = \dfrac{\text{nærliggende katet}}{\text{hypotenus}} = \dfrac{cot(\alpha)}{csc(\alpha)} = \dfrac{1}{sec(\alpha)}$

$\,$

$\Large tan(\alpha) = \dfrac{\text{motsatt katet}}{\text{nærliggende katet}} = \dfrac{sin(\alpha)}{cos(\alpha)} = \dfrac{1}{cot(\alpha)}$

$\,$

$\Large sec(\alpha) = \dfrac{\text{hypotenus}}{\text{nærliggende katet}} = \dfrac{csc(\alpha)}{cot(\alpha)} = \dfrac{1}{cos(\alpha)}$

$\,$

$\Large cot(\alpha) = \dfrac{\text{nærliggende katet}}{\text{motsatt katet}} = \dfrac{cos(\alpha)}{sin(\alpha)} = \dfrac{1}{tan(\alpha)}$

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$\Large csc(\alpha) = \dfrac{\text{hypotenus}}{\text{motsatt katet}} = \dfrac{sec(\alpha)}{tan(\alpha)} = \dfrac{1}{sin(\alpha)}$

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$\,$

$\Large sin^2 (\alpha) + cos^2 (\alpha) = 1$

$\,$

$\Large tan(\alpha) = \dfrac{sin(\alpha)}{cos(\alpha)}$

$\,$

$\Large tan(\alpha) \cdot cos(\alpha) = 1$

$\,$

$\Large sin(2\alpha) = 2 sin(\alpha) \cdot cos(\alpha)$

$\,$

$\Large cos(2\alpha) = cos^2(\alpha) - sin^2(\alpha) = 2 cos^2(\alpha) -1$

$\,$

$\Large cos(2\alpha) = 1 - 2 sin^2(\alpha)$

$\,$

$\Large tan(2\alpha) = \dfrac{2 tan(\alpha)}{1-tan^2(\alpha)}$

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$\Large sin(3 \alpha) = 3 sin(\alpha) - 4 sin^3(\alpha)$

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$\Large cos(3 \alpha) = 4 cos^3(\alpha) - 3 cos(\alpha)$

$\,$

$\Large sin(\alpha ± \beta) = sin(\alpha) \cdot cos(\beta) ± cos(\alpha) \cdot sin(\beta)$

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$\Large cos(\alpha + \beta) = cos(\alpha) \cdot cos(\beta) - sin(\alpha) \cdot sin(\beta)$

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$\Large cos(\alpha - \beta) = cos(\alpha) \cdot cos(\beta) + sin(\alpha) \cdot sin(\beta)$

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$\Large tan(\alpha + \beta) = \dfrac{tan(\alpha) + tan(\beta)}{1 - tan(\alpha) \cdot tan(\beta)}$

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$\Large tan(\alpha - \beta) = \dfrac{tan(\alpha) - tan(\beta)}{1 + tan(\alpha) \cdot tan(\beta)}$

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$\Large sin(\alpha) + sin(\beta) = 2 sin(\dfrac{\alpha + \beta}{2}) \cdot cos(\dfrac{\alpha + \beta}{2})$

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$\Large sin(\alpha) - sin(\beta) = 2 cos(\dfrac{\alpha + \beta}{2}) \cdot sin(\dfrac{\alpha + \beta}{2})$

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$\Large cos(\alpha) + cos(\beta) = 2 cos(\dfrac{\alpha + \beta}{2}) \cdot cos(\dfrac{\alpha + \beta}{2})$

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$\Large cos(\alpha) - cos(\beta) = -2 sin(\dfrac{\alpha + \beta}{2}) \cdot sin(\dfrac{\alpha + \beta}{2})$

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$\Large sin(\alpha) \cdot cos(\beta) = \dfrac{1}{2} \left( sin(\alpha + \beta) + sin(\alpha - \beta) \right)$

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$\Large sin(\beta) \cdot cos(\alpha) = \dfrac{1}{2} \left( sin(\alpha + \beta) - sin(\alpha - \beta) \right)$

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$\Large cos(\beta) \cdot cos(\alpha) = \dfrac{1}{2} \left( cos(\alpha + \beta) + cos(\alpha - \beta) \right)$

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$\Large sin(\beta) \cdot sin(\alpha) = \dfrac{1}{2} \left( cos(\alpha - \beta) - cos(\alpha + \beta) \right)$