Chapter 17
Compound angles
17.1 Compound angle formulae
An electric current i may be expressed as i =
5 sin(ωt − 0.33) amperes. Similarly, the displacement
x of a body from a fixed point can be expressed as
x = 10 sin(2t + 0.67) metres. The angles (ωt − 0.33) and
(2t + 0.67) are called compound angles because they
are the sum or difference of two angles. The compound
angle formulae for sines and cosines of the sum and
difference of two angles A and B are:
sin(A + B) = sin A cos B + cos A sin B
sin(A − B) = sin A cos B − cos A sin B
cos(A + B) = cos A cos B − sin A sin B
cos(A − B) = cos A cos B + sin A sin B
(Note, sin(A + B) is not equal to (sin A + sin B), and
so on.)
The formulae stated above may be used to derive two
further compound angle formulae:
tan(A + B) =
tan A + tan B
1 − tan A tan B
tan(A − B) =
tan A − tan B
1 + tan A tan B
The compound-angle formulae are true for all values of
A and B, and by substituting values of A and B into 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) = sin π cos α + cos π sin α (from
the formula forsin(A + B))
= (0)(cos α) + (−1) sin α = −sin α
(b) −cos(90 ◦ + β)
= −[cos 90
◦ cos β − sin 90
◦ sin β]
= −[(0)(cos β) − (1) sin β] = sin β
(c) sin(A − B) − sin(A + B)
= [sin A cos B − cos A sin B]
− [sin A cos B + cos A sin B]
= −2cos A sin B
Problem 2. Prove that
cos(y − π) + sin
y +
π
2
= 0.
cos(y − π) = cos y cos π + sin y sin π
= (cos y)(−1) + (sin y)(0)
= −cos y
sin
y +
π
2
= sin y cos
π
2
+ cos y sin
π
2
= (sin y)(0) + (cos y)(1) = cos y
Hence
cos(y − π)+ sin
y +
π
2
= (−cos y) + (cos y) = 0
Problem 3. Show that
tan
x +
π
4
tan
x −
π
4
= −1.