Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

1.2 TRIGONOMETRIC IDENTITIES


x

y

x′

y′

O


A


B


P


T


N


R


M


Figure 1.2 Illustration of the compound-angle identities. Refer to the main
text for details.

Other standard single-angle formulae derived from (1.15) by dividing through

by various powers of sinθand cosθare


1+tan^2 θ=sec^2 θ, (1.16)
cot^2 θ+1=cosec^2 θ. (1.17)

1.2.2 Compound-angle identities

The basis for building expressions for the sinusoidal functions of compound


angles are those for the sum and difference of just two angles, since all other


cases can be built up from these, in principle. Later we will see that a study of


complex numbers can provide a more efficient approach in some cases.


To prove the basic formulae for the sine and cosine of a compound angle

A+Bin terms of the sines and cosines ofAandB, we consider the construction


shown in figure 1.2. It shows two sets of axes,OxyandOx′y′, with a common


origin but rotated with respect to each other through an angleA. The point


Plies on the unit circle centred on the common originOand has coordinates


cos(A+B),sin(A+B) with respect to the axesOxyand coordinates cosB,sinB


with respect to the axesOx′y′.


Parallels to the axesOxy(dotted lines) andOx′y′(broken lines) have been

drawn throughP. Further parallels (MRandRN)totheOx′y′axes have been

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