1.2 TRIGONOMETRIC IDENTITIES
xyx′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 throughby 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 identitiesThe 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 angleA+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 beendrawn throughP. Further parallels (MRandRN)totheOx′y′axes have been