1 Trigonometry Samuel Marateck © 2009

2 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.

3 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.

4 An identity is an equations that is true for all values of the unknown variable. An example of an identity is cos 2 (x) + sin 2 (x) = 1. We will be using the identity: e ix = 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. e i0 = cos(0) + i sin(0) so 1 = 1.

5 We will multiply e ix = cos(x) + i sin(x) by e iy = cos(y) + i sin(y) so e ix e iy =(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 i 2 = -1. Or e ix e iy =cos(x) cos(y)-sin(x)sin(y)+i[sin(x)cos(y)+cos(x) sin(y)] But e ix e iy =e i(x+y) = cos(x+y) + i sin(x + y). We equate the real parts and the imaginary ones of e ix e iy and e i(x+y).

6 (1)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) = ½ [cos(x+y) + cos(x-y) ] The factor of ½ makes sense since the maximum value of the left side is 1.0. Without the ½ the maximum value of right side would be 2.0.

7 We see from cos(x) cos(y) = ½ [cos(x+y) + cos(x-y) ] that multiplication on the left side is equated to addition on the right side. This is reminiscent of logarithms. Let’s see if we can use this to multiply two numbers. Let’s first evaluate cos(x) cos(y). We will express the numbers we wish to multiply in scientific notation so we get a number less than 1.0

8 We multiply by The result Is or x In order to converts these numbers to cosines, we express these numbers as x 10 6 and x 10 6 respectively. In a number like x 10 6, the part is the mantissa. Using the mantissa, we see that the angle whose cos is is 44 o and the angle whose cos is is 29 0.

9 Now using these angles, the right side of cos(x) cos(y) = ½ [cos(x+y) + cos(x-y) ] becomes ½[cos( ) + cos( )] or ½[cos(73 0 ) + cos(15 0 )]. Looking up these two cosines, we get: ½ [ ] = When we multiplied the original two numbers we got x to six significant figures.

10 The reason we got such good correspondence is that we were simply verifying the trigonometry identity since the numbers we multiplied corresponded to exact angles. In general we would have to find the closest angle corresponding to the cosines of the numbers we would like to multiply. The closer to an angle in the cosine table we come, the better the approximation.