where e
ix
= cos(
x
) +
i
sin(
x
), sometimes abbreviated to cis(
x
).
When this substitution of
t
for tan(
x
/2) is used in calculus, it follows that sin(
x
) is replaced by 2
t
/(1 +
t
2
),
cos(
x
) is replaced by (1
−
t
2
)/(1 +
t
2
) and the differential
dx
is replaced by (2
dt
)/(1 +
t
2
). Thereby one
converts rational functions of sin(
x
) and cos(
x
) to rational functions of
t
in order to find their
antiderivatives.
See also
Trigonometry
Proofs of trigonometric identities
Uses of trigonometry
Tangent half-angle formula
Law of cosines
Law of sines
Law of tangents
Half-side formula
Mollweide's formula
Pythagorean theorem
Exact trigonometric constants (values of sine and cosine expressed in surds)
Derivatives of trigonometric functions
List of integrals of trigonometric functions
Hyperbolic function
Prosthaphaeresis
Versine and haversine
Exsecant
Notes
1.
^
Abramowitz and Stegun, p. 73, 4.3.45
2.
^
Abramowitz and Stegun, p. 78, 4.3.147
3.
^
Abramowitz and Stegun, p. 72, 4.3.13–15
4.
^
The Elementary Identities ( )
5.
^
Abramowitz and Stegun, p. 72, 4.3.9
6.
^
Abramowitz and Stegun, p. 72, 4.3.7–8
7.
^
Abramowitz and Stegun, p. 72, 4.3.16
8.
^
a
b
c
Weisstein, Eric W., "Trigonometric Addition Formulas
( )" from MathWorld.
9.
^
Abramowitz and Stegun, p. 72, 4.3.17
10.
^
Abramowitz and Stegun, p. 72, 4.3.18
11.
^
Abramowitz and Stegun, p. 80, 4.4.42
12.
^
Abramowitz and Stegun, p. 80, 4.4.43
13.
^
Abramowitz and Stegun, p. 80, 4.4.36
14.
^
Bronstein, Manual (1989). "Simplification of Real Elementary Functions".
Proceedings of the ACM-
SIGSAM 1989 international symposium on Symbolic and algebraic computation
: 211.
15.
^
a
b
Weisstein, Eric W., "Multiple-Angle Formulas ( -
AngleFormulas.html)" from MathWorld.
Page 24 of 25
List of trigonometric identities - Wikipedia, the free encyclopedia
10/4/2011