Math 2.0

Special functions Theorem

Angle addition formulas

Builds on Sine and cosineCauchy product of absolutely convergent series

Statement

For all real numbers xx and yy, sin⁡(x+y)=sin⁡xcos⁡y+cos⁡xsin⁡y,\sin(x + y) = \sin x \cos y + \cos x \sin y, cos⁡(x+y)=cos⁡xcos⁡y−sin⁡xsin⁡y.\cos(x + y) = \cos x \cos y - \sin x \sin y.

Remarks

One proof multiplies the power series with the Cauchy product and uses the binomial theorem. Another proof fixes yy and shows that the difference of the two sides solves f′′=−ff'' = -f with f(0)=f′(0)=0f(0) = f'(0) = 0. With y=xy = x, they give sin⁡2x=2sin⁡xcos⁡x\sin 2x = 2 \sin x \cos x and cos⁡2x=cos⁡2x−sin⁡2x\cos 2x = \cos^2 x - \sin^2 x. Mathlib states them as Real.sin_add and Real.cos_add.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionCauchy product, definition, no evidence levelCauchy productdefinitionPartial sum, definition, no evidence levelPartial sumdefinitionPower series, definition, no evidence levelPower seriesdefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionSine and cosine, definition, no evidence levelSine and cosinedefinitionAbsolute convergence, definition, no evidence levelAbsolute convergencedefinitionCauchy product of absolutely convergent series, statement, no evidence levelCauchy product of abs…statementAngle addition formulas, statement, no evidence levelAngle addition formul…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Sequence definition
  2. Real sequence definition
  3. Limit of a sequence definition
  4. Series definition
  5. Convergent sequence definition
  6. Cauchy product definition
  7. Partial sum definition
  8. Power series definition
  9. Convergent series definition
  10. Sine and cosine definition
  11. Absolute convergence definition
  12. Cauchy product of absolutely convergent series statement
  13. Angle addition formulas statement

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/List_of_trigonometric_identities, Angle sum and difference identities
About this page
Tag028XEQCE88
URIm2://028XEQCE88
Kindstatement, theorem
WritingW0 stub
MSC33B10
StatusThis copy of the site has no build data. Run m2 status 028XEQCE88.
UsesSine and cosine, Cauchy product of absolutely convergent series

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