Higher Engineering Mathematics

COMPOUND ANGLES 185

B

From equation (7),

cos 6x+cos 2x=2 cos 4xcos 2x

From equation (5),

sin 6x+sin 2x=2 sin 4xcos 2x

Hence

cos 6x+cos 2x

sin 6x+sin 2x

=

2 cos 4xcos 2x

2 sin 4xcos 2x

=

cos 4x

sin 4x

=cot 4x

Now try the following exercise.

Exercise 84 Further problems on changing

sums or differences of sines and cosines into

products

In Problems 1 to 5, express as products:

1. sin 3x+sinx [2 sin 2xcosx]
   2.^12 (sin 9θ−sin 7θ) [cos 8θsinθ]


2. cos 5t+cos 3t [2 cos 4tcost]
   4.^18 (cos 5t−cost)

[

−^14 sin 3tsin 2t

]

5.^12

(

cos

π

3

+cos

π

4

) [

cos

7 π

24

cos

π

24

]

6. Show that:

(a)

sin 4x−sin 2x

cos 4x+cos 2x

=tanx

(b)^12 {sin(5x−α)−sin(x+α)}

=cos 3xsin(2x−α)

18.6 Power waveforms in a.c. circuits

(a) Purely resistive a.c. circuits

Let a voltagev=Vmsinωtbe applied to a circuit
comprising resistance only. The resulting current is
i=Imsinωt, and the corresponding instantaneous
power,p, is given by:

p=vi=(Vmsinωt)(Imsinωt)

i.e., p=VmImsin^2 ωt

From double angle formulae of Section 18.3,

cos 2A= 1 −2 sin^2 A, from which,

sin^2 A=^12 (1−cos 2A) thus

sin^2 ωt=^12 (1−cos 2ωt)

Then power p=VmIm[^12 (l−cos 2ωt)]

i.e. p=^12 VmIm(1−cos 2ωt)

The waveforms ofv,iandpare shown in Fig. 18.8.

The waveform of power repeats itself afterπ/ω

seconds and hence the power has a frequency twice

that of voltage and current. The power is always pos-

itive, having a maximum value ofVmIm. The average

or mean value of the power is^12 VmIm.

Figure 18.8

The rms value of voltageV= 0. 707 Vm, i.e.V=

Vm

√

2

,

from which,Vm=

√

2 V.

Similarly, the rms value of current,I=

Im

√

2

, from

which, Im=

√

2 I. Hence the average power, P,

developed in a purely resistive a.c. circuit is given

byP= 21 VmIm=^12 (

√

2 V)(

√

2 I)=VIwatts.

Also, powerP=I^2 RorV^2 /Ras for a d.c. circuit,

sinceV=IR.

Summarizing, the average powerPin a purely

resistive a.c. circuit given by

P=VI=I^2 R=

V^2

R

whereVandIare rms values.

(b) Purely inductive a.c. circuits

Let a voltagev=Vmsinωtbe applied to a circuit

containing pure inductance (theoretical case). The