How to Prove the Trigonometric Identities for Sin and Cos of A+B?

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SUMMARY

The trigonometric identities for the sine and cosine of the sum of two angles, specifically sin(A+B) = sin A cos B + cos A sin B and cos(A+B) = cos A cos B - sin A sin B, are fundamental concepts in trigonometry. These identities can be proven through various methods, which are documented in standard trigonometry textbooks and online resources. A recommended starting point for these proofs is the Wikipedia page on trigonometric identities.

PREREQUISITES
  • Basic understanding of trigonometric functions (sine, cosine)
  • Familiarity with angle addition formulas
  • Knowledge of mathematical proof techniques
  • Access to trigonometry textbooks or reliable online resources
NEXT STEPS
  • Study the proofs of trigonometric identities in standard trigonometry textbooks
  • Explore the Wikipedia page on Proofs of Trigonometric Identities
  • Practice deriving other trigonometric identities using angle addition formulas
  • Learn about the applications of trigonometric identities in physics and engineering
USEFUL FOR

Students of mathematics, educators teaching trigonometry, and anyone seeking to deepen their understanding of trigonometric identities and their proofs.

basty
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Why

##\sin (A+B) = \sin A \cos B + \cos A \sin B##

and

##\cos (A+B) = \cos A \cos B - \sin A \sin B##

??
 
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The proof of this should be available in any basic textbook on trigonometry or by a simple search in a search engine. I suggest starting here: http://en.wikipedia.org/wiki/Proofs_of_trigonometric_identities
Please note that Physics Forums is not a substitute for a textbook or intended to spoon feed you basic mathematical facts.
 

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