Difference between revisions of "Angle addition identities"
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label("$\frac{1}{\cos \beta}$",D--O,dir(325)); | label("$\frac{1}{\cos \beta}$",D--O,dir(325)); | ||
</asy> | </asy> | ||
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+ | <math>\fontsize{18}{27}\selectfont \sin (\alpha + \beta ) = \frac{\left( \sin \alpha + \frac{\cos \alpha \sin \beta}{\cos \beta} \right)}{\frac{1}{\cos \beta}} = \cos \beta \times \left( \sin \alpha + \frac{\cos \alpha \sin \beta}{\cos \beta} \right) = \sin \alpha \cos \beta + \cos \alpha \sin \beta</math> | ||
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+ | <math>\fontsize{18}{27}\selectfont \cos (\alpha + \beta ) = \frac{\left( \cos \alpha - \frac{\sin \alpha \sin \beta}{\cos \beta} \right)}{\frac{1}{\cos \beta}} = \cos \beta \times \left( \cos \alpha - \frac{\sin \alpha \sin \beta}{\cos \beta} \right) = \cos \alpha \cos \beta - \sin \alpha \sin \beta</math> | ||
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+ | <math>\fontsize{18}{27}\selectfont \tan (\alpha + \beta ) = \frac{\sin (\alpha + \beta )}{\cos (\alpha + \beta )} = \frac{\sin \alpha \cos \beta + \cos \alpha \sin \beta}{\cos \alpha \cos \beta - \sin \alpha \sin \beta} = \frac{\frac{\sin \alpha \cos \beta + \cos \alpha \sin \beta}{\cos \alpha \cos \beta}}{\frac{\cos \alpha \cos \beta - \sin \alpha \sin \beta}{\cos \alpha \cos \beta}} = \frac{\frac{\sin \alpha}{\cos \alpha} + \frac{\sin \beta}{\cos \beta}}{1 - \frac{\sin \alpha \sin \beta}{\cos \alpha \cos \beta}} = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}</math> | ||
==See Also== | ==See Also== | ||
* [[Trigonometric identities]] | * [[Trigonometric identities]] |
Latest revision as of 19:46, 13 January 2024
The trigonometric angle addition identities state the following identities:
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Proofs