Mathematical Methods for Physics and Engineering - Matematica.NET

1.2 TRIGONOMETRIC IDENTITIESy ′yRPMTNBx ′OAxFigure 1.2 Illustration of the compound-angle identities. Refer to the maintext for details.Other standard single-angle formulae derived from (1.15) by dividing throughby various powers of sin θ and cos θ are1+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 compoundangles are those for the sum and difference of just two angles, since all othercases can be built up from these, in principle. Later we will see that a study ofcomplex numbers can provide a more efficient approach in some cases.To prove the basic formulae for the sine and cosine of a compound angleA + B in terms of the sines and cosines of A and B, we consider the constructionshown in figure 1.2. It shows two sets of axes, Oxy and Ox ′ y ′ , with a commonorigin but rotated with respect to each other through an angle A. The pointP lies on the unit circle centred on the common origin O and has coordinatescos(A + B), sin(A + B) with respect to the axes Oxy and coordinates cos B,sin Bwith respect to the axes Ox ′ y ′ .Parallels to the axes Oxy (dotted lines) and Ox ′ y ′ (broken lines) have beendrawn through P . Further parallels (MR and RN) totheOx ′ y ′ axes have been11