There’s something truly beautiful about the internal consistencies of mathematics. Thanks to its nature we get to hike on such a variety of paths to reach the same destination. Math becomes a mental traveler’s paradise.
Consider the trigonometric addition formulas:
sin(x + y) = sin(y) cos(x) + sin(x) cos(y) and
cos (x + y) = cos (x) cos(y) – sin(x) sin(y) .
These lead to highly useful double angle formulas, which are needed to evaluate integrals. They are also used in Fourier analysis and in communication systems. How do we derive them? I’ll first explore a slightly more difficult route. Secondly we will briefly refer to a conventional geometrical approach and finally we’ll examine my own offbeat path (but I can’t guarantee it’s original) .
(1) A circuitous route using slightly higher mathematics
Knowing the rate of change of sine, cosine and natural base functions allows one to express sin(w), cos(w) and ew as power series. Replace the w with xi, where i is the square root of -1, the so-called imaginary number, and exi turns out to be the sum of the cosine series and i times the sine series, namely Euler’s relationship:
exi = cos (x) + i sin(x).
If we subsequently use the above to express ei(x+y) we obtain
ei(x+y) = cos (x + y) + i sin(x + y). equation(1)
But using exponent laws, ei(x+y) = exi (eyi ) , and then applying Euler’s formula again:
ei(x+y) = exi [eyi ] = [cos(x) + i sin(x) ] [cos(y) + i sin(y) ] and expanding the right hand side:
ei(x+y) = cos(x) cos(y) + i sin(y) cos(x) + i sin(x) cos(y) – sin(x) sin(y).
If we regroup the above by pairing and separating the real components from the imaginary ones :
ei(x+y) = cos(x) cos(y) – sin(x) sin(y) + i [sin(y) cos (x) + sin(x) cos (y)]. equation(2)
we can compare equations (1) and (2) and have to conclude that:
cos(x + y) = cos(x) cos(y) – sin(x) sin(y).
sin(x + y) = sin(y) cos(x) + sin(x) cos(y).
(2) A Geometric Approach
Elementary math can be used if we begin with the following diagram. The details of this particular derivation are found at the site indicated in the caption.
(3) A simpler off beat path from working backwards
After deriving the Sine Law from simple trig ratios, I subsequently thought of using the same diagram to make substitutions into the sine addition formula. Surely enough, after some manipulations, I obtained the Sine Law, implying that the latter can be used to derive the other.
By using the same approach with the cosine addition formula, I obtained another unexpected starting point. Apologies for not typing it out. The convenience of simply photographing my handwritten versions proved to be too tempting!