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.