Understanding Engineering Mathematics

B. (i) sin(A−B)=sin(A+(−B))

=sinAcos(−B)+sin(−B)cosA

=sinAcosB−sinBcosA

on using

cos(−x)=cosxand sin(−x)=−sinx

(ii) The result for cos(A−B)may now be obtained using comple-

mentary angles (149

➤

):

cos(A−B)=sin( 90 −(A−B))

=sin( 90 −A+B)

=sin( 90 −A)cosB+cos( 90 −A)sinB

=cosAcosB+sinAsinB

Or, perhaps quicker, simply differentiate (➤233) the result for

sin(A−B)with respect toB– a common trick for obtaining ‘cos-

results’ from ‘sin-results’.

(iii) The result for tan(A+B)can be obtained by division:

tan(A+B)=

sin(A+B)

cos(A+B)

=

sinAcosB+sinBcosA

cosAcosB−sinAsinB

=

tanA+tanB

1 −tanAtanB

on dividing top and bottom by cosAcosB.

The key point here is that whilecertainresults have to be second

nature to you, such as the expansion of sin(A+B),itisjustas

important that you know how to derive others from them, rather

than have to remember all the identities.

C.Here we make good use of the compound angle formulae

(i) cos 75°=cos( 45 °+ 30 °)

=cos 45°cos 30°−sin 45°sin 30°

=

1

√

2

√

3

2

−

1

√

2

1

2

=(

√

3 − 1 )

√

2

4

(ii) sin 105°=sin( 60 °+ 45 °)

=sin 60°cos 45°+sin 45°cos 60°

=

√

3

2

1

√

2

+

1

√

2

1

2

=

√

3 + 1

2

√

2

=

(

√

3 + 1 )

√

2

4

(iii) tan(− 75 °)=−tan 75°

=−tan( 45 °+ 30 °)