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Mathematical Equivalence of Analytic and Geometric
Definitions of Sine and Cosine Functions
Abstract
In high school trigonometry we learn the geometric definitions of the sine and
cosine functions as
sine = perpendicular/hypotenuse, cosine = base/hypotenuse.
Later, usually in college calculus, we learn their analytic definitions as
cos x = (eix + e-ix)/2, sin x = (eix - e-ix)/2i
The paper establishes the mathematical equivalence of these two definitions.
1.
Analytic Definitions Imply Geometric Definitions
(Adapted from Walter Rudin – Principles of Mathematical Analysis.)
The proof consists of the following steps:
(1.1) Define functions
C(t) = (eit + e-it)/2, S(t) = (eit - e-it)/2i
We use C(t) and S(t) instead of cos t and sin t respectively to avoid any possible
use of the geometric definitions of the sine and cosine functions.
2)
Prove that there exist positive real numbers x such that C(x) = 0. Let x0 be
the smallest positive real number such that C(x0) = 0. Define a positive
real number λ by λ = 2x0. Then C(λ/2) = 0, S(λ/2) = 1, and ez+2λi = ez.
Hence ez is periodic with period 2λi, and C(x) and S(x) are periodic with
period 2λ.
3)
Prove that if 0 < t < 2 λ and e4it is real, then eit ≠ 1.
4)
Prove that if |z| = 1, then there exists a unique real number t, 0 ≤ t < 2λ
such that eit = z.
5)
Define a curve η by η(t) = eit, 0 ≤ t < 2λ. Then, η is a closed curve whose
range is the unit circle and whose length is 2 λ. Hence λ = π. Thus, C(t)
and S(t) are periodic functions with period 2 π and ez is periodic with
period 2πi.
2
6)
As t increases from 0 to t0, η(t) describes a circular arc subtending an
angle t0 at the origin O (0,0). Consider the triangle with vertices O = (0,0),
A = (C(t0), 0), and P = (C(t0), S(t0)). We get
C(t0) = OA/OP = cos t0, and S(t0) = AP/OP = sin t0,
which are the geometric definitions of cosine and sine functions.
2.
Geometric Definitions Imply Analytic Definitions
The proof consists of the following steps:
(2.1) Start with the definitions
sine = perpendicular/hypotenuse, cosine = base/hypotenuse.
2)
Prove (see any calculus book) that
lim sin h/h = 1, lim (cos h – 1)/h = 0.
h→0
h→0
3)
Using (2.2) prove that (sin x)’ = cos x, (cos x)’ = - sin x.
(2.4) Take a point z on the unit circle |z| = 1. Let the polar coordinates of z be
(1, θ). Since z = x + iy, we get x = cos θ and y = sin θ. Hence we get,
z = cos θ + i sin θ, and dz/dθ = - sin θ + i cos θ.
Integrating we get Euler’s formula,
eiθ = cos θ + i sin θ,
whence we derive the analytic definitions of sine and cosine functions
. cos θ = (eiθ + e-iθ)/2, sin θ = (eiθ - e-iθ)/2i