Difference between revisions of "Angle addition identities"

(Created page with "The trigonometric angle addition identities state the following identities: <math>\sin(x + y) = \sin (x) \cos (y) + \cos (x) \sin (y)</math> <math>\cos(x + y) = \cos (x) \cos...")
 
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<math>\cos(x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)</math>
 
<math>\cos(x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)</math>
 
<math>\tan(x + y) = \frac{\tan (x) + \tan (y)}{1 - \tan (x) \tan (y)}</math>
 
<math>\tan(x + y) = \frac{\tan (x) + \tan (y)}{1 - \tan (x) \tan (y)}</math>
There are many proofs of these identities. For the sake of brevity, we list only one here.
 
 
Euler's identity states that <math>e^{ix} = \cos (x) + i \sin(x)</math>. We have thatcos(x+y)+isin(x+y)=ei(x+y)=eixeiy=(cos(x)+isin(x))(cos(y)+isin(y))=(cos(x)cos(y)sin(x)sin(y))+i(sin(x)cos(y)+cos(x)sin(y))By looking at the real and imaginary parts, we derive the sine and cosine angle addition formulas.
 
 
To derive the tangent addition formula, we reduce the problem to use sine and cosine, divide both numerator and denominator by <math>\cos (x) \cos (y)</math>, and simplify.tan(x+y)=sin(x+y)cos(x+y)=sin(x)cos(y)+cos(x)sin(y)cos(x)cos(y)sin(x)sin(y)=sin(x)cos(x)+sin(y)cos(y)1sin(x)sin(y)cos(x)cos(y)=tan(x)+tan(y)1tan(x)tan(y)as desired.
 
  
 
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Revision as of 19:20, 7 September 2023

The trigonometric angle addition identities state the following identities:

$\sin(x + y) = \sin (x) \cos (y) + \cos (x) \sin (y)$ $\cos(x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)$ $\tan(x + y) = \frac{\tan (x) + \tan (y)}{1 - \tan (x) \tan (y)}$

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