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Weierstrass substitution formulas∗
Wkbj79†
2013-03-21 22:53:08
The Weierstrass substitution formulas for −π < x < π are:
sin x =
2t
1 + t2
cos x =
1 − t2
1 + t2
dx
2
dt
1 + t2
=
They can be obtained in the following manner:
x
Make the Weierstrass substitution t = tan
. (This substitution is also
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known as the universal trigonometric substitution.) Then we have
cos
x
2
=
sec
1
x
2
1
=r
x
1 + tan2
2
=√
1
1 + t2
and
sin
x
2
= cos
=√
x
2
· tan
x
2
t
.
1 + t2
∗ hWeierstrassSubstitutionFormulasi
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Note that these are just the “formulas involving radicals” as designated in
the entry goniometric formulas; however, due to the restriction on x, the ±’s
are unnecessary.
Using the above formulas along with the double angle formulas, we obtain
x
x
· cos
sin x = 2 sin
2
2
=2· √
=
t
1
·√
1 + t2
1 + t2
2t
1 + t2
and
cos x
= cos2
=
x
2
− sin2
1
√
1 + t2
2
x
−
=
t2
1
−
1 + t2
1 + t2
=
1 − t2
.
1 + t2
2
t
√
1 + t2
2
x
Finally, since t = tan
, solving for x yields that x = 2 arctan t. Thus,
2
2
dt.
dx =
1 + t2
The Weierstrass substitution formulas are most useful for integrating rational
functions of sine and cosine.
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