Chapter 17

Compound angles

17.1 Compound angle formulae

An electric current i maybeexpressedasi=
5sin(ωt− 0. 33 )amperes. Similarly, the displacement
xof a body from a fixed point can be expressed as
x=10sin( 2 t+ 0. 67 )metres.Theangles(ωt− 0 .33)and
(2t+ 0 .67) are calledcompound anglesbecause they
are the sum or difference of two angles. Thecompound
angle formulaefor sines and cosines of the sum and
difference of two anglesAandBare:

sin(A+B)=sinAcosB+cosAsinB

sin(A−B)=sinAcosB−cosAsinB

cos(A+B)=cosAcosB−sinAsinB

cos(A−B)=cosAcosB+sinAsinB

(Note, sin(A+B)isnotequal to(sinA+sinB),and
so on.)
The formulae stated above may be used to derive two
further compound angle formulae:

tan(A+B)=

tanA+tanB

1 −tanAtanB

tan(A−B)=

tanA−tanB

1 +tanAtanB

The compound-angle formulae are true for all values of
AandB, and by substitutingvalues ofAandBinto the
formulae they may be shown to be true.

Problem 1. Expand and simplify the following

expressions:

(a) sin(π+α) (b)−cos( 90 ◦+β)

(c) sin(A−B)−sin(A+B)

(a) sin(π+α)=sinπcosα+cosπsinα(from

the formula forsin(A+B))

=( 0 )(cosα)+(− 1 )sinα=−sinα

(b) −cos( 90 ◦+β)

=−[cos90◦cosβ−sin90◦sinβ]

=−[( 0 )(cosβ)−( 1 )sinβ]=sinβ

(c) sin(A−B)−sin(A+B)

=[sinAcosB−cosAsinB]

−[sinAcosB+cosAsinB]

=−2cosAsinB

Problem 2. Prove that

cos(y−π)+sin

(

y+

π

2

)

= 0.

cos(y−π)=cosycosπ+sinysinπ

=(cosy)(− 1 )+(siny)( 0 )

=−cosy

sin

(

y+

π

2

)

=sinycos

π

2

+cosysin

π

2

=(siny)( 0 )+(cosy)( 1 )=cosy

Hence cos(y−π)+sin

(

y+

π

2

)

=(−cosy)+(cosy)= 0

Problem 3. Show that

tan

(

x+

π

4

)

tan

(

x−

π

4

)

=− 1.