Q&A: What Does it Mean to Say that a Proof is Trigonometric or Calculus-Based?

Cross-posted from here.

What does it mean to say that a proof is trigonometric or calculus-based? Can’t any proof ultimately be reduced down to algebra and logic?


When we say that a proof is “trigonometric,” all we’re saying is that it uses trigonometric functions. That’s it.

Sure, you could write trigonometric proofs without using trigonometric functions. You could reduce everything down to simple algebra, explicitly writing $b/c$ in place of $\sin A.$

But does that mean it’s not trigonometric? No.

Otherwise, you could apply the same reasoning to argue ridiculous things, e.g. that the Fundamental Theorem of Calculus

$$\begin{align*} \int_a^b F'(x) \, \textrm dx = F(b) - F(a) \end{align*}$$

is not actually differential/integral calculus because you can reduce it down to just limits:

$$\begin{align*} \lim_{n \to 0} \sum_{i=0}^{n-1} \left( \lim_{h \to 0} \frac{F(x_i +h)-F(x_i)}{h} \right) \Delta x = F(b) - F(a) \end{align*}$$

where $\Delta x = \dfrac{b-a}{n}$ and $x_i = a + i \Delta x.$

So, when we say the proof is trigonometric, we mean that it uses trigonometric notation to describe symbolic patterns leveraged in trigonometry, not that it is impossible to state without trigonometric notation.