Understanding Engineering Mathematics

A Q

A

R

P

O

B

A

ST

Figure 6.13Proof of sin(A+B)=sinAcosB+cosAsinB.

PuttingA=B in the compound angle identities immediately gives thedouble angle
formulae:
sin 2A≡sinAcosA
cos 2A≡cos^2 −sin^2 A
≡2cos^2 A− 1
≡ 1 −2sin^2 A

tanA≡

2tanA

1 −tan^2 A

The cos 2Aresults are often more useful in the form

sin^2 A=^12 ( 1 −cos 2A)

cos^2 A=^12 ( 1 +cos 2A)

From the double angle formulae we easily deduce thehalf angle identities.Ift=tanθ/2,
then:

tanθ≡

2 t

1 −t^2

sinθ≡

2 t

1 +t^2

cosθ≡

1 −t^2

1 +t^2

For example:

tanθ=tan( 2 ×θ/ 2 )=

2tanθ/ 2

1 −tan^2 θ/ 2

=

2 t

1 −t^2

from the double angle formula.

Solution to review question 6.1.7

A.sin(A+B)=sinAcosB+sinBcosAis so important it should be at

your fingertips.