cs.nyu.educoursesfall09v55.0109-001trigonometry.ppt-.ppt

Trigonometry Samuel Marateck ? 2009 How some trigonometric identities presaged logarithms. We will be using complex numbers to simplify derivations. A complex number consists of a real part and an imaginary one. An imaginary number is one followed or preceded by the imaginary unit i, where i is √(-1). Examples are: 123.5i, i345, ui and iv. An example of a complex number is s = u + iv. An example of another one is t = w + ix. If two complex numbers, here s and t are equal, their real parts must be equal, and their imaginary parts must be equal. So here u = w and v = x. An identity is an equations that is true for all values of the unknown variable. An example of an identity is cos2(x) + sin2(x) = 1. We will be using the identity: eix = cos(x) + i sin(x) where x is an angle. We know that sin(0) equals 0 and cos(0) is 1. So we can check the above formula by setting x to 0. ei0 = cos(0) + i sin(0) so 1 = 1. We will multiply eix = cos(x) + i sin(x) by eiy = cos(y) + i sin(y) so eix eiy =(cos(x) + i sin(x) )(cos(y) + i sin(y) ) = cos(x) cos(y) + i sin(x) cos(y) + i cos(x) sin(y) - sin(x) sin(y) since i2 = -1. Or eix eiy=cos(x) cos(y)-sin(x)sin(y)+i[sin(x)cos(y)+cos(x) sin(y)] But eix eiy=ei(x+y) = cos(x+y) + i sin(x + y). We equate the real parts and the imaginary ones of eix eiy and ei(x+y). cos(x+y) = cos(x) cos(y)-sin(x) sin(y) for the real part and (2) sin(x+y) = sin(x) cos(y)+cos(x) sin(y) for the imaginary part. We will use the first equation. Since the sin(-y) = -sin(y) and the cos(-x) = cos(x), we have cos(x-y) = cos(x) cos(-y)-sin(x) sin(-y) or (3) cos(x-y) = cos(x) cos(y)+sin(x) sin(y), adding (1) and (3) (4) cos(x+y) + cos(x-y) = 2 cos(x) cos(y) or (5) cos(x) cos(y) = ? [cos(x+y) + cos(x-y) ] The factor of ? makes sense since the maximum value of the left side is 1.0. Without the ? the maximum value of right side would be 2.0. We see from cos(x) cos(y) = ? [cos(x+y) + cos(x-y) ] that multiplication on the left side is equated to addition o