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Trigonometric Series Formulas

Trigonometric identities which prove useful in the construction of map projections include

Asin(2phi)+Bsin(4phi)+Csin(6phi)+Dsin(8phi) =sin(2phi)(A^'+cos(2phi)(B^'+cos(2phi)(C^'+D^'cos(2phi)))),
(1)

where

A^'=A-C
(2)
B^'=2B-4D
(3)
C^'=4C
(4)
D^'=8D.
(5)
Asinphi+Bsin(3phi)+Csin(5phi)+Dsin(7phi) =sinphi(A^'+sin^2phi(B^'+sin^2phi(C^'+D^'sin^2phi))),
(6)

where

A^'=A+3B+5C+7D
(7)
B^'=-4B-20C-56D
(8)
C^'=16C+112D
(9)
D^'=-64D.
(10)
A+Bcos(2phi)+Ccos(4phi)+Dcos(6phi)+Ecos(8phi) =A^'+cos(2phi)(B^'+cos(2phi)(C^'+cos(2phi)(D^'+E^'cos(2phi)))),
(11)

where

A^'=A-C+E
(12)
B^'=B-3D
(13)
C^'=2C-8E
(14)
D^'=4D
(15)
E^'=8E
(16)

(Snyder 1987).

See also

Trigonometry

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References

Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, p. 19, 1987.

Referenced on Wolfram|Alpha

Trigonometric Series Formulas

Cite this as:

Weisstein, Eric W. "Trigonometric Series Formulas." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TrigonometricSeriesFormulas.html

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