Quantum Physics

A.5 Trigonometry A.7When natural logarithms are used,y ln e x (or x e y )[A.15]For example, log 10 52 1.716, so that antilog 10 1.716 10 1.716 52. Likewise,ln e 52 3.951, so antiln e 3.951 e 3.951 52.In general, note that you can convert between base 10 and base e with theequalityln e x (2.302 585)log 10 x[A.16]Finally, some useful properties of logarithms arelog (ab) log a log blog (a/b) log a log blog (an ) n log aln e 1ln e a aln 1a ln aA.4 GEOMETRYTable A.2 gives the areas and volumes for several geometric shapes used throughoutthis text:TABLE A.2Useful Information for GeometrywArea = wrSurface area = 4pr 2Volume = 4 3 pr 3RectangleSphererArea = pr 2Circumference = 2prrSurface area = 2pr 2 + 2prVolume = pr 2 CircleCylinderbTriangleh1 Area = bh 2hRectangular boxwSurface area =2(h + w + hw)Volume = whA.5 TRIGONOMETRYSome of the most basic facts concerning trigonometry are presented in Chapter 1,and we encourage you to study the material presented there if you are having troublewith this branch of mathematics. In addition to the discussion of Chapter 1,certain useful trig identities that can be of value to you follow.sin 2 cos 2 1sin cos(90 )cos sin(90 )sin 2 2 sin cos cos 2 cos 2 sin 2 sin( ) sin cos cos sin cos( ) cos cos sin sin