Approximating sin(0.1)

Real Analysis · Axiom Academy

Your calculator has no sin circuit — only +, −, ×, ÷. So how does it return sin(0.1) to ten correct digits in an instant? The trick is a polynomial: sin x = x − x³/6 + x⁵/120 − … Keep a few terms, plug in x = 0.1 , and the answer falls out of pure arithmetic. Three moves: stack the terms , read the error , see why small x is the whole game . Add Maclaurin terms one at a time and watch the running sum close in on the true value. Each new term is smaller than the last, and the answer ping-pongs over the target by less and less. Read the error before you compute it Here is the real-analysis payoff: you can bound the error without knowing the true value. Because the series alternates, the error is never bigger than the first term you threw away. Drag and watch the guarantee tighten. Why small x is the whole trick Keep just two terms and slide the angle. At x = 0.1 the error is a speck; push x up and it erupts — the leftover grows like x⁵. That is why fast routines first shrink the angle, then approximate. Three moves, one idea: stack a few terms , let the next term bound the error , and keep the angle small . The same recipe — reduce, then sum a short polynomial — is how every calculator and CPU computes sin , cos , eˣ , ln , and the rest.

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