Dictionary:Trigonometric identities

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Some statements which hold for all angles are shown in Figure T-15.

Trigonometric Identities

Euler's relation

  • From complex analysis we learn that a complex number
  • is called the modulus
  • is the called the phase

Complex exponential form of sine and cosine

Double angle formulas

Cosine and Sine of a sum and difference of angles

Products of Cosines and Sines

Deriving the identities

We can show that it is entirely plausible that the identities above are true.

Euler's relation


We begin by formally writing the Taylor series representations of and and sum the resulting series



where We may write the Taylor series form of


where . In both of the previous cases, free use has been made of the identity

The sum of these two series yields the series representation of the exponential function


where .

This is not a proof of Euler's relation. It is only an argument of plausibility, because we have not proven the many theorems that would allow us to establish the existence of the complex exponential, the Taylor series, or its convergence. Rarely will \thetas argument of plausibility appear in a textbook on complex analysis, owing to its heuristic nature.

Complex exponential form of trigonometric functions

Given that

and that, owing to the eveness of cosine and the oddness of sine,

we may write the complex conjugate of as

.

The complex exponential form of sine and cosine follow from the respective sum and difference of and

The other identities derive from the complex exponential form of sine and cosine.