From Euclid to Spigotry #4
Trigonometric Tools
Last modified: September 15, 2004
Since p can be defined in terms of properties of the circle,  relationships among  the 
"circular functions" (sine, cosine, tangent, etc.) -- such things as addition formulas and 
double-angle formulas -- are important.  Euler's discovery that the exponential of an 
imaginary number can be expressed in terms in sines and cosines placed the exponential 
function in the same category. 
This collection of short lectures should present all the trigonometric tools that we will need in order to analyze Greek problems in modern terms.

Reference:
Bold, Chapter III

Lecture Topic 1: Review of trigonometry:
Given a right triangle as in the diagram to the right,
with c = , here are the trigonometric functions.
Sine sin q = a/c
Cosine   cos q = b/c
Tangent     tan q = sin q /cos q = a/b
Cotangent cot q = cos q / sin q = b/a
Secant: sec q = 1/ cos q = c/b 
Cosecant:     csc q = 1 / sin q = c/a
Starting from the Pythagorean theorem c2 = a2 + b2 , prove three useful identities.
Divide by c2 : 1 = sin2 q + cos2 q
Divide by b2:  sec2 q = tan2 q + 1
Divide by a2:  csc2 q = 1 + cot2 q 

By looking at the angle b that equals p/2 - q,
show that 
sin b = sin (p/2 - q) = cos q
cos b = cos (p/2 - q) = sin q

In terms of coordinate geometry, explain why it is reasonable for trig functions to be defined for negative angles by

cos (-q) = cos q
sin (-q) = -sin q

Lecture topic 2: The addition formulas
It takes an argument involving congruent triangles to get started.

If the circle has unit radius, the coordinates of A are (cos a, sin a)
The coordinates of B are (cos b, sin b)
The coordinates of Q are (cos (a-b), sin (a-b))
The coordinates of P are (1, 0).
Triangles OAB and OPQ are congruent, so the distance AB equals the distance QP.
In algebraic terms, this says that
(cos a - cos b)2 + (sin a - sin b)2 = (cos (a - b) - 1)2 + sin2 (a - b).

Multiply this all out, and you find that 
cos (a - b) = cos a cos b + sin a sin b
Replace b by -b and you find that
cos (a + b) = cos a cos b - sin a sin b
Replace a by p/2 - a, and you find that
cos (p/2 - a - b) = cos (p/2 - a) cos b + sin (p/2 - a) sin b
which means that
sin (a + b) = sin a cos b + cos a sin b

Lecture topic 3: De Moivre's theorem
This states that (cos q + i sin q)m = cos mq + i sin mq
The classic proof by induction is on page 22 of Bold
Lecture topic 4: Euler's complex exponential function
Euler defined e iq = cos q + i sin q.
Now from the rule for multiplying exponentials, 
you can rederive the following:
1. cos2 q + sin2 q =1.
2. sin(a + b) = sin a cos b + cos a sin b
3. cos(a + b) = cos a cos b - sin a sin b
4. De Moivre's theorem:
This is done on pp. 21-22 of Bold, but without the use of Euler's exponential notation
Lecture topic 5: Multiple-angle formulas
To understand why the Greeks could construct a regular pentagon but not a regular heptagon with compass and straightedge, we need to be able to express sin nx or cos nx in terms of powers of sin x and cos x. Computer algebra programs like Mathematica or Mathcad (used for this document) can do this automatically.
Examples: 

Show how easy it is to obtain these by using De Moivre's theorem, for example
cos 2x + i sin 2x = (cos x + i sin x)2
Just expand the right-hand side, equate the real parts and imaginary parts, and you get two formulas for the price of one!   Do the same with
cos 3x + i sin 3x = (cos x + i sin x)3
Lecture topic 6: Half-angle formulas
Archimedes had the basic idea about how to "square the circle" by repeatedly bisecting angles to 
obtain polygons with more and more sides.  The trigonometric equivalent of his approach is to find 
formulas for the sines and cosines of q/2.  Starting from

cos q = 2 cos2 (q/2) - 1 = 1 - 2 sin2 (q/2),
find formulas for cos (q/2), sin (q/2) and tan (q/2) in terms of cos q.
Lecture topic 7: Addition and double-angle formulas for tangents and cotangents:
Machin's formula, which was used in 2002 to calculate 1.24 trillion digits of p, and Richmond's
famous 19th century construction of the pentagon, both rely on formulas involving tangents and cotangents.  Prove that
tan (x + y) =  and that
cot(2x) = (cot x - tan x)/2

Lecture topic 8: Special values
It's obvious from an isosceles right triangle that 
sin (p/4) = cos(p/4) =  
Starting from this, use the double-angle formulas
sin 2q = 2 sin q cos q 
and
cos 2q = cos2 q - sin2 q
to determine the sine and cosine of p/2, p, and 2p.
Use the addition formulas to show that 
sin(q + p) = - sin q
and sin (q + 2p) = sin q.
Draw diagrams to show that all these results make sense once you know that the coordinates
of a point on the unit circle are (cos q, sin q) no matter how large the angle q may be.