Understanding Engineering Mathematics

so

cos^2 θ=

2

3

cosθ=

√

2

3

sin^2 θ= 1 −cos^2 θ= 1 −

2

3

=

1

3

and therefore

sinθ=

1

√

3

6.2.7 Compound angle formulae

➤

173 196➤

The basic compound angle formulae are:

sin(A+B)≡sinAcosB+cosAsinB

sin(A−B)≡sinAcosB−cosAsinB

cos(A+B)≡cosAcosB−sinAsinB

cos(A−B)≡cosAcosB+sinAsinB

tan(A+B)≡

tanA+tanB

1 −tanAtanB

tan(A−B)≡

tanA−tanB

1 +tanAtanB

Note that there is no need to remember all these results separately. We can get all of
them from the result for sin(A+B), and the elementary properties of the trig ratios (see
solution to review question). Also, get used to ‘knowing them backwards’ that is going
from right to left, recognising how the right-hand side simplifies to the left-hand side.
Since the sin(A+B)result is so important we will take the trouble to prove it. The proof
is instructive since it contains lots of ideas already covered.
We use one of those mystical constructions which makes geometry so pretty. Consider
Figure 6.13.
PQis drawn atQperpendicular to the line making angleAwith the base lineOT,and
meets the corresponding line defining the compound angleA+BatP.PSandQT are
dropped perpendicular toOTandRQis then drawn parallel toOT. By alternate angles
(149

➤
) OQR=A.SincePQRis a right angled triangleQPRis the complement
(149

➤

)ofRQPand is therefore equal toOQR=QOT=A.Also,RS=QT.

Now sin(A+B)=

PS

OP

=

PR+RS

OP

=

PR+QT

OP

=

PR

OP

+

QT

OP

=

QT

OQ

·

OQ

OP

+

PR

QP

·

QP

OP

=sinAcosB+cosAsinB