الهويات المثلثية

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This page is a translated version of the page Dictionary:Trigonometric identities and the translation is 5% complete.
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بعض العبارات التي تحمل جميع الزوايا موضحة في الشكل T-15.

Trigonometric Identities

Euler's relation

  • From complex analysis we learn that a complex number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z = x + i y = r (\cos \theta + i \sin \theta) = r e^{i \theta} }
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r = \sqrt{ x^2 + y^2 } } is called the modulus
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta = \arctan(y/x) } is the called the phase

Complex exponential form of sine and cosine

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cos \theta = \frac{e^{i\theta} + e^{-i \theta}}{2} }
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sin \theta = \frac{e^{i\theta} - e^{-i \theta}}{2i} }

Double angle formulas

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cos^2 \theta = \frac{1}{2} \left[ 1 + \cos 2 \theta \right] }
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sin^2 \theta = \frac{1}{2} \left[ 1 - \cos 2 \theta \right] }

Cosine and Sine of a sum and difference of angles

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cos( \theta_1 \pm \theta_2 ) = \cos \theta_1 \cos \theta_2 \mp \sin \theta_1 \sin \theta_2 }
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sin ( \theta_1 \pm \theta_2 ) = \sin \theta_1 \cos \theta_2 \pm \sin \theta_2 \cos \theta_1 }

Products of Cosines and Sines

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cos \theta_1 \cos \theta_2 = \frac{1}{2} \left[ \cos(\theta_1 + \theta_2) - \cos(\theta_1 - \theta_2) \right] }
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sin \theta_1 \sin \theta_2 = \frac{1}{2} \left[ \cos(\theta_1 - \theta_2) - \cos(\theta_1 + \theta_2) \right] }
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sin \theta_1 \cos \theta_2 = \frac{1}{2} \left[ \sin(\theta_1 + \theta_2) + \sin(\theta_1 - \theta_2) \right] }

Deriving the identities

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

Euler's relation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cos(\theta) + i \sin(\theta) = e^{i \theta}. }


We begin by formally writing the Taylor series representations of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cos (\theta) } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i \sin (\theta) } and sum the resulting series


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cos (\theta) = \sum_{k=0}^{\infty} \frac{(-1)^{k} \theta^{2k} }{ (2k)! } = \sum_{k=0}^{\infty} \frac{(i)^{2k} \theta^{2k} }{ (2k)! } = \sum_{k=0}^{\infty} \frac{( i \theta)^{2k} }{ (2k)! }, }


where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k=0,1,2,3,... } We may write the Taylor series form of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i \sin(\theta) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i \sin (\theta) = i \sum_{l=0}^{\infty} \frac{(-1)^{l} \theta^{2l+1} }{ (2l+1)! } = \sum_{l=0}^{\infty} \frac{(i)^{2l+1} \theta^{2l+1} }{ (2l+1)! } = \sum_{l=0}^{\infty} \frac{( i \theta)^{2l+1} }{ (2l+1)! } ,}


where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle l=0,1,2,3,...} . In both of the previous cases, free use has been made of the identity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -1 = i^2.}

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


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cos(\theta) + i \sin (\theta) = \sum_{n=0}^{\infty} \frac{( i \theta)^{n} }{ n! } = e^{ i \theta }, }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n = 0,1,2,3,... } .

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle e^{i \theta} = \cos(\theta) + i \sin(\theta) }

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

we may write the complex conjugate of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \exp(i\theta) } as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle e^{-i \theta} = \cos(\theta) - i \sin(\theta) }

.

The complex exponential form of sine and cosine follow from the respective sum and difference of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \exp(i \theta) } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \exp(-i \theta). }

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