Download Trigonometry

Trigonometry
Samuel Marateck © 2009
1
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.
2
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.
3
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.
4
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).
5
(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.
6
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
7
We multiply 719340 by 874620. The result
Is 629149150800 or 0.629149 x 1012.
In order to converts these numbers to cosines, we
express these numbers as .719340 x 106 and
.874620 x 106 respectively. In a number like
.719340 x 106, the .719340 part is the mantissa.
Using the mantissa, we see that the angle whose
cos is .719340 is 44o and the angle whose cos is
.874620 is 290.
8
Now using these angles, the right side of
cos(x) cos(y) = ½ [cos(x+y) + cos(x-y) ] becomes
½[cos(440+290) + cos(440-290)] or
½[cos(730) + cos(150)]. Looking up these
two cosines, we get:
½ [.292372 + .965926] = .629149
When we multiplied the original two
numbers we got 0.629149 x 1012 to
six significant figures.
9
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.
10