Mathematical Methods for Physics and Engineering - Matematica.NET

PRELIMINARY ALGEBRAdrawn through R, the point (0, sin(A + B)) in the Oxy system. That all the anglesmarked with the symbol • are equal to A follows from the simple geometry ofright-angled triangles and crossing lines.We now determine the coordinates of P in terms of lengths in the figure,expressing those lengths in terms of both sets of coordinates:(i) cos B = x ′ = TN + NP = MR + NP= OR sin A + RP cos A = sin(A + B)sinA +cos(A + B)cosA;(ii) sin B = y ′ = OM − TM = OM − NR= OR cos A − RP sin A = sin(A + B)cosA − cos(A + B)sinA.Now, if equation (i) is multiplied by sin A and added to equation (ii) multipliedby cos A, the result issin A cos B +cosA sin B = sin(A + B)(sin 2 A +cos 2 A)=sin(A + B).Similarly, if equation (ii) is multiplied by sin A and subtracted from equation (i)multiplied by cos A, the result iscos A cos B − sin A sin B =cos(A + B)(cos 2 A +sin 2 A)=cos(A + B).Corresponding graphically based results can be derived for the sines and cosinesof the difference of two angles; however, they are more easily obtained by settingB to −B in the previous results and remembering that sin B becomes − sin Bwhilst cos B is unchanged. The four results may be summarised bysin(A ± B) =sinA cos B ± cos A sin B (1.18)cos(A ± B) =cosA cos B ∓ sin A sin B. (1.19)Standard results can be deduced from these by setting one of the two anglesequal to π or to π/2:sin(π − θ) =sinθ, cos(π − θ) =− cos θ, (1.20)sin ( 12 π − θ) =cosθ, cos ( 12 π − θ) =sinθ, (1.21)From these basic results many more can be derived. An immediate deduction,obtained by taking the ratio of the two equations (1.18) and (1.19) and thendividing both the numerator and denominator of this ratio by cos A cos B, istan A ± tan Btan(A ± B) =1 ∓ tan A tan B . (1.22)One application of this result is a test for whether two lines on a graphare orthogonal (perpendicular); more generally, it determines the angle betweenthem. The standard notation for a straight-line graph is y = mx + c, inwhichmis the slope of the graph and c is its intercept on the y-axis. It should be notedthat the slope m is also the tangent of the angle the line makes with the x-axis.12