Trigonometry
The simplest way to introduce Trigonometric functions is to begin with their unit circle definitions, and then specialize to right triangle cases with the help of some similarity properties.
Step 1.
Draw a unit circle
Your unit of measurement may be one centimeter, one meter, one kilometer, one inch, one foot, one yard, one mile or any other unit. Choose one, or draw a circle and declare its radius to be your unit length.
Step 2.
Let q be an angle. Locate the head of the vector with angle q and length 1 on the unit circle.
Step 3.
The head will have coordinates (a units, b units)
Put cos(q) =a and sin (q) =b. This defines both sine and cosine for all values of the angle q.
Further trig functions may be defined as follows.
when the divisors are nonzero.
The case where q is between 0 and 90 degrees is considered next.
Step 4 (Right Triangle Trigonometry)
Assume q is between 0 and 90 degrees. Then
For angles between 0 and 90 degrees, similarity of right triangles implies the ratios
if you replace the unit circle right triangle by a similar right triangle.
The latter formulas for may be used to compute 
with any right triangle where sides are labeled
opposite and adjacent for an angle 
The further trig functions may be defined as follows.
when the divisors are nonzero.
Exercise: Express these further trig functions as ratios of the sides opposite, adjacent and/or hypotenuse of the above right triangle.
Calculation
One may define trig functions by saying how to compute them in principle as above, but then one computes or approximates them in practice from tables and slide rules (old fashioned approach) or using calculators (the new approach). Unfortunately in this practice, the tables, slide rules or calculation devices are black boxes which provide results, but whose derivation or justification is not commonly known. This departs from the principle of understanding the computations one does, but the numbers computed by these black boxes can be checked in simple cases. When calculators first arrived, some used faulty or suboptimal methods (algorithms) to compute.
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 trig course will explain the following in more detail.
Trig functions link the ratio of two sides of a right triangle to cosines, sines and tangents of an angle. Knowledge of two sides in right triangle gives knowledge of the third by means of Pythagorean theorem, and of the values of the trig functions for the angles in the triangle. Computation of unknown side lengths, unknown hypotenuse lengths and unknown angles is useful in land measurement (geo - metry) and also in navigation.
From one-to-one properties of trig functions for angles between 0 and 90 degrees or ½p, one can define (say how to compute) inverse trig functions to compute the angles from the ratio of sides. Computation with inverse trig functions allows one to obtain polar coordinates for vectors or complex numbers from coordinates, real and imaginary parts, or the length of the adjacent and opposite sides of a right triangle determined by the coordinates. Again, this removes the need to measure the lengths and angles for points with rectangular coordinates [a, b].
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Complex Numbers |
The fundamental theorem of algebra and partial fraction decomposition in calculus depend on complex numbers.
Easy Consequences
Hint: See the (newest) Complex
Number. Starter Lesson.
for a simple geometric introduction, then continue with easy consequence below.
The clearest geometric proof of the distributive law appears in the
folder.
First Earlier (Old) exposition of complex numbers follows in Z1 to B1 below - read for review or revision .
D1 to D6 after provide a review of vectors.
More on Complex Numbers:
Chapters in Volume 3::
Intro assumes the field properties
of real numbers in place of
deriving them geometrically