Ask a calculus student what sin x means, and you'll almost certainly hear "opposite over hypotenuse." It's a fine answer for a right triangle with an acute angle in it. It is a completely useless answer for the question a calculus course actually needs answered: what is sin(−4.7), and why is its derivative cos(−4.7)? SOH-CAH-TOA has nothing to say about negative numbers, or numbers bigger than π/2, or why the resulting function should be differentiable at all. Michael Spivak opens Chapter 15 of Calculus by admitting exactly this: "The definitions of the functions sin and cos are considerably more subtle than one might suspect." What follows is one of the most satisfying rebuilding jobs in the whole book.
Where the Triangle Definition Quietly Breaks
The trouble isn't that the triangle picture is wrong — it's that it's radically incomplete. A right triangle only has room for angles strictly between 0° and 90°. The moment you want sin(150°) or sin(−30°), "opposite over hypotenuse" has already run out of triangle. Textbooks patch this with extra rules about signs in different quadrants, but that's a patch, not a foundation. And even if you accept the patched-together function, nothing about it tells you it's differentiable, which is the one property calculus actually needs. Spivak wants sin and cos to be genuine functions on all of R, built from definitions precise enough to prove things about — not a lookup table of triangle ratios with quadrant rules bolted on.
First, You Have to Define the Angle Itself
Spivak's fix starts further back than you'd expect: before defining sine, you have to pin down what an angle is, as a number. He sets this up using directed angles — pairs of half-lines from a common point — and observes that if you always fix one half-line along the positive horizontal axis, a directed angle is completely described by where the other half-line crosses the unit circle: a point (x, y) with x² + y² = 1. That's already an improvement, since every real point on the circle now corresponds to some angle, all the way around and back again.
But you still need to attach a number to that angle, and here Spivak flags a genuinely subtle issue most courses paper over: "degrees" require you to already know what "90 degrees" or "√2 degrees" means, which is circular. The fix is radian measure: given a number x, walk counterclockwise along the unit circle starting at (1, 0) a distance of exactly x, and call the point you land on "the angle of x radians." This is the first real payoff of thinking in arc length instead of protractor markings — it defines an angle for every real number, positive or negative, automatically wrapping around every 2π.
The Actual Definition: Area, Not Length
Arc length is intuitive, but Spivak points out it's technically awkward to work with at this stage (defining the length of a curve rigorously takes real machinery), so he reformulates the whole thing in terms of area, which the book has already nailed down via integrals. He starts by pinning down π itself as a number, with no appeal to circles-are-round hand-waving:
π = 2·∫−11 √(1−x²) dx
That is: π is defined to be twice the area of the upper unit semicircle. Next comes the function that does all the real work: for −1 ≤ x ≤ 1, let A(x) be the area of the circular sector between the positive horizontal axis and the half-line through the point (x, √(1−x²)). Working out the area of a triangle plus the area under a curve gives the explicit formula
A(x) = [x·√(1−x²)] / 2 + ∫x1 √(1−t²) dt
Now the definition of cosine and sine, for 0 ≤ x ≤ π, is exactly the reverse of what you'd guess from the triangle picture: cos x is defined as the unique number in [−1, 1] for which A(cos x) = x/2, and then sin x is defined as √(1 − (cos x)²). Instead of starting with an angle and asking for a ratio, you start by demanding a sector of a specific area, x/2, and cos x is whatever number makes that true. Proving such a number exists at all requires an appeal to the Intermediate Value Theorem, since A is continuous and decreases from A(−1) = π/2 down to A(1) = 0. Only after this does Spivak extend the definition beyond [0, π] with an explicit piecewise rule using 2π-periodicity, finally producing functions defined on the whole real line.
The Payoff: The Derivatives Fall Out for Free
Here's why all that machinery earns its keep. Because cos is now literally defined as an inverse function (of B(x) = 2A(x)), Spivak can differentiate it using the inverse function rule, and it costs almost nothing:
A′(x) = −1 / (2√(1−x²)) ⇒ cos′(x) = −√(1−(cos x)²) = −sin x
and then, differentiating sin x = √(1−(cos x)²) directly, sin′(x) = cos x drops out in three lines. Nothing here was assumed — it was proved, using nothing but the Fundamental Theorem of Calculus and the inverse function rule, from a definition built entirely out of integrals.
A Concrete Consequence: The Addition Formula, for Free
The most striking payoff comes a few pages later. Spivak proves a lemma: if f has a second derivative everywhere and satisfies f″ + f = 0 with f(0) = f′(0) = 0, then f is identically zero — by noticing that (f′)² + f² is constant and equal to 0. From this one lemma, he shows that any function satisfying f″ + f = 0 is completely determined by f(0) and f′(0): specifically f = f′(0)·sin + f(0)·cos. Then, fixing any number y and defining f(x) = sin(x + y), you can check f″ + f = 0 with f(0) = sin y and f′(0) = cos y — and the theorem above hands you, with essentially no trigonometric manipulation at all, the addition formula sin(x + y) = sin x cos y + cos x sin y. A formula generations of students memorize from a picture of two stacked triangles turns out to be a one-paragraph consequence of a differential equation.
This is the real argument for building sine and cosine this way: the definition isn't more rigorous just to satisfy a pedant, it's more productive. Once sin and cos are honest functions built from areas and integrals, their derivatives, their extension to all of R, and their addition formulas all become short proofs instead of things you're asked to accept on faith. If you want to see the full argument — including how Spivak handles the boundary cases and previews an even faster route through arcsin in Chapter 18 — it's laid out in complete detail in Spivak's Calculus, 4th Edition.
Further reading: How to self-study Spivak's Calculus · Table of contents · Index · Calculus, 4th Edition · Answer Book
Which definition would you rather build a whole theory on: the one with a triangle, or the one with an integral?
