Chapter 42
The t = tan
θ
2
substitution
42.1 Introduction
Integrals of the form
∫
1
acosθ+bsinθ+c
dθ,where
a,bandcare constants, may be determined by usingthe
substitutiont=tan
θ
2
. The reason is explained below.
If angleAin the right-angled triangleABCshown in
Fig. 42.1 is made equal to
θ
2
then, since tangent=
opposite
adjacent
,ifBC=tandAB=1, then tan
θ
2
=t.
By Pythagoras’ theorem,AC=
√
1 +t^2
A^2
1
B
11 t^2
C
t
Figure 42.1
Therefore sin
θ
2
=
t
√
1 +t^2
and cos
θ
2
=
1
√
1 +t^2
Since
sin2x=2sinxcosx (from double angle formulae,
Chapter 17), then
sinθ=2sin
θ
2
cos
θ
2
= 2
(
t
√
1 +t^2
)(
t
√
1 +t^2
)
i.e. sinθ=
2 t
( 1 +t^2 )
(1)
Since cos2x=cos^2
θ
2
−sin^2
θ
2
=
(
1
√
1 +t^2
) 2
−
(
t
√
1 +t^2
) 2
i.e. cosθ=
1 −t^2
1 +t^2
(2)
Also, sincet=tan
θ
2
,
dt
dθ
=
1
2
sec^2
θ
2
=
1
2
(
1 +tan^2
θ
2
)
from trigonometric
identities,
i.e.
dt
dθ
=
1
2
( 1 +t^2 )
from which, dθ=
2dt
1 +t^2
(3)
Equations (1), (2) and (3) are used to determine
integrals of the form
∫
1
acosθ+bsinθ+c
dθwhere
a,borcmay be zero.