Unit Circle and Right Triangle Developments of Trigonometry
A. Trigonometry
The simplest way to introduce trigonometric functions (functions on your
calculator) is to begin with their unit circle definitions, and then
specialize to their right triangle computation with the help of
similarity assumptions about triangles, right or scalene. Several steps
follow for reading in or besides your trig course.
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.
Question: How does similarity assumptions for
right triangles imply the results, here the definition of trig
functions below, is independent of the choose of 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)
on circle of radius 1 unit.
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)
circle of radius 1 unit.
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.
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 (more
functions on your calculator) 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].
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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