Understanding Engineering Mathematics

t = −aw

0 t

A

2 p

w

Figure 17.6Sinusoidal functionAsin(ωt+α).

Exercise on 17.7

Show that (i) the sum of two sinusoids with the same frequency.

Asin(ωt)+Bsin(ωt+α)

and (ii) the integral of a sinusoid

∫

asin(ωt+φ)dt

each have the same frequency as the original sinusoids, and determine the amplitude and
phase of the result in each case.

Answer

(i) Amplitude is

√

(A+Bcosα)^2 +Bsin^2 α

Phase isβwhere tanβ=

Bsinα

A+Bcosα

(ii) Amplitude is

a

ω

. Phase is shifted by

π

2

17.8 Orthogonality relations for trigonometric functions

In the process of expressing a general waveform of a given period in terms of sinusoids of
given frequency – the object of Fourier analysis – we will need certain integral formulae.
These are so important that we devote this whole section to them. Make sure that you
fully understand what they mean, and can derive them (285
➤
). We will only consider
integrals and sinusoids defined over a period of 2π, usually taken to be−πtoπ.Thisis
no real restriction – if we have a region of different length then we can simply scale it to
2 π. Our integrals concern products of sines and cosines over the range−πtoπ.