Trigonometry and Complex Numbers
If z = (r,q) in polar
coordinates, then z = a + i b =[a,b] = [r
cos(q), r sin(q)]
in rectangular coordinates. So the ability to compute
cosines and sines avoids the need to measure the rectangular
coordinates after a diagram after locating the point z from
its polar coordinates.
A quick way, or the quickest way, to
understand and fully explain the algebraic properties
of trig functions is online is to start a trig
courses after this site treatment of complex complex
numbers
The following webpages in the earlier composed complex
number site area demonstrate how a knowledge of complex numbers,
may aid and enrich trigonometry.
| B7 Dot & Cross
Products |
See the link between the rectangular coordinate
computation of dot and cross products
[x1,y1].[x2,y2]
= x1x2+y1y2 (dot product
definition)
[x1,y1]×[x2,y2]
= x1y2 - y1x2
(cross product definition)
for vectors or points [x1,y1] =
and [x2,y2] in the plane and the trig functions
cosine and sine. |
| B8 Cosine Law |
Here are proofs
of the cosine law and a converse to the Pythagorean theorem. |
| B9 Exponential &
cis functions |
The formula
cis(q) = cos(q)+isin(q)
= exp(iq) and properties of
complex numbers (two ways to multiply) imply Cosine and Sine Addition Formulas. |
| B10 Easy Trig
Identities |
The formula
cis(q) = cos(q)+isin(q)
= exp(iq) and properties of
complex numbers (two ways to multiply) yield further trig identities for
cos(nA) and sin (nA) in the case n =2 and 3. (From the
binomial theorem for the expansion of (a+b)n follow formula for
the general case.. |
| Chapter
24 in Volume 3 Why Slopes and More Math ) |
Logs, Powers and Exponentials of Complex Numbers
Preview of Electrical Engineering Mathematics |
| Pages
B&, B8 and B9 can be read in any order. Page B10 has B9 as a
prerequisite. |
Some other pages in the complex
number site area are part of chains of reason out of sequence with those in
this site area.
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