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.