8 Trigonometric
Trigonometric functions
8.1 Trigonometric functions and their derivatives We are study-
ing and perhaps even learning mathematics, and it may be amusing and
perhaps even useful to see how an unscientific but highly critical justice
of a Supreme Court might be introduced to angles and trigonometry.
We would teach him enough about numbers, E2j continuous functions,
and curves to make him realize that if we start with positive numbers a
and h for which 0 < h < a, then the ordered set of points (x, a2 - x2)
for which h x S a, the point for which x = x2 preceding the point
for which x = xiL when a <- xl < x2 < h, is a curve. Our curve lies on
the circle with center at the origin and radius a, and we simplify (or
complicate) matters by calling our curve "the arc, of the circle with
center at the origin and radius a, which runs in the positive direction
from the point (a,0) to the point Q having coordinates (h, a2 --h2)"
or, for temporary purposes, simply "the arc." To help us remember the
meanings of our symbols, we can start constructing Figure 8.11 and
amplify it as we proceed. Next we must teach our pupil a theory of
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