Mathematical Methods for Physics and Engineering - Matematica.NET

1.2 TRIGONOMETRIC IDENTITIESConsequently the angle θ 12 between two such straight-line graphs is equal to thedifference in the angles they individually make with the x-axis, and the tangentof that angle is given by (1.22):tan θ 12 = tan θ 1 − tan θ 21+tanθ 1 tan θ 2= m 1 − m 21+m 1 m 2. (1.23)For the lines to be orthogonal we must have θ 12 = π/2, i.e. the final fraction onthe RHS of the above equation must equal ∞, andsom 1 m 2 = −1. (1.24)A kind of inversion of equations (1.18) and (1.19) enables the sum or differenceof two sines or cosines to be expressed as the product of two sinusoids; theprocedure is typified by the following. Adding together the expressions given by(1.18) for sin(A + B) and sin(A − B) yieldssin(A + B)+sin(A − B) =2sinA cos B.If we now write A + B = C and A − B = D, this becomes( ) C + Dsin C +sinD =2sin cos2( C − DIn a similar way each of the following equations can be derived:2). (1.25)( ) ( )C + D C − Dsin C − sin D =2cos sin , (1.26)22( ) ( )C + D C − Dcos C +cosD =2cos cos , (1.27)22( ) ( )C + D C − Dcos C − cos D = −2sin sin . (1.28)22The minus sign on the right of the last of these equations should be noted; it mayhelp to avoid overlooking this ‘oddity’ to recall that if C>Dthen cos C