In trigonometry and related fields, it's common knowledge that the sine and cosine functions are intimately connected with the unit circle. It's often taken for granted that these functions describe the x and y coordinates on the circle as it's parametrized by an angle θ. The equations are typically given as:
$$ x(\theta) = \cos(\theta) $$ $$ y(\theta) = \sin(\theta) $$
where $( \theta )$ is the angle from the positive x-axis to the radius of the circle in a counter-clockwise direction.
I'm looking for a formal mathematical proof or explanation that clearly shows how these functions relate to the geometric circle. Specifically, I'm interested in understanding the foundational principles that establish sine and cosine as the x and y coordinates of a circle when the angle θ is varied.
Can anyone provide an explanation or point me toward resources that prove or visually demonstrate this relationship beyond just the unit circle definition?
The problem reduces to proving that $~\sin^2(\theta) + \cos^2(\theta) = 1,~$ since :
In $~\Bbb{R^2},~$ the unit circle equals $~\{(x,y) \;: \;x^2 + y^2 = 1\}.$
For a circle of radius $~r,~$ any point $~(x,y)~$ on the circle will satisfy $~x = r\cos(\theta), \;y = r\sin(\theta).$
The method of proving this depends on whether the context is [trigonometry:plane geometry] or real analysis.
$\underline{\text{In Trigonometry - Plane Geometry}}$
By the pythagorean theorem, the three sides of a right triangle satisfy $~x^2 + y^2 = c^2,~$ where $~x,y~$ are the horizontal and vertical legs, respectively, and $~c~$ is the hypotenuse.
Then, the issue is resolved by the definition of sine and cosine as opposite over hypotenuse and adjacent over hypotenuse respectively.
$\underline{\text{In Real Analysis}}$
Here I will excerpt the approach in "Calculus" 2nd Ed, Vol 1, 1966, by Tom Apostol.
Apostol defines the sine and cosine functions via the following axioms (if such functions might exist):
A1 : The sine and cosine functions are defined for all $x$ in $\mathbb{R}.$
A2: $\cos (0) = \sin(\pi/2) = 1, \cos(\pi) = -1.$
A3: $\cos(y - x) = \cos y \cos x + \sin y \sin x.$
A4: For $0 < x < \pi/2, ~0 < \cos x < \frac{\sin x}{x} < \frac{1}{\cos x}.$
Paraphrasing Apostol, he then makes the following point:
Examination of the traditional sine and cosine functions against the backdrop of the unit circle (centered at the origin) demonstrates that functions that satisfy the axioms do exist, as long as the domain of the functions is construed to be the dimensionless length of the corresponding arc (rather than an angle).
Then, based solely on the axioms, Apostol proves that $~\sin^2(\theta) + \cos^2(\theta) = 1,~$ as follows:
Invoke A2 and A3, setting $~y = x = \theta.~$
This implies that
$$1 = \cos(0) = \cos(\theta - \theta) = \cos^2(\theta) + \sin^2(\theta).$$