关于使sin(x)为超越数的规整x取值及典型值判定的技术咨询

Hey there! First off, no worries about being new here—we’ve all been there, and your question is a really interesting deep dive into transcendental number theory, so great job framing it.

Let’s break this down step by step, starting with the specific examples you asked about:

First, is sin(1 radian) really transcendental? #

Absolutely, and we can prove it using the Lindemann-Weierstrass Theorem—a foundational result in transcendental number theory. This theorem states that if α is a non-zero algebraic number, then e^α is transcendental.

Here’s how it applies to sin(1):
Using Euler’s formula, we know sin(x) = (e^(ix) - e^(-ix))/(2i). Suppose for contradiction that sin(1) were algebraic. Then rearranging the formula gives us a quadratic equation in e^i:
(e^i)^2 - 2i sin(1) e^i - 1 = 0
This would mean e^i is a root of a polynomial with algebraic coefficients, making e^i algebraic—but 1 is a non-zero algebraic number, so i (algebraic) times 1 is still algebraic, and Lindemann-Weierstrass tells us e^i must be transcendental. Contradiction! So sin(1) has to be transcendental. That’s why Wolfram Alpha can state it confidently.

"Nice" values of x where sin(x) is confirmed to be transcendental #

Based on the same theorem and related results, we can confirm transcendence for several categories of "nice" x:

• Non-zero rational numbers in radians: If q ∈ ℚ and q ≠ 0, then sin(q) is transcendental. The proof is identical to the sin(1) case: iq is a non-zero algebraic number, so e^(iq) is transcendental, and assuming sin(q) is algebraic leads to a contradiction.
• Algebraic irrational numbers in radians: For example, x = √2, x = Φ = (1+√5)/2 (the golden ratio), or x = √3. Since these are algebraic numbers (non-zero), ix is also algebraic, so e^(ix) is transcendental. Using the same Euler formula trick as before, we can show sin(x) can’t be algebraic—so it’s transcendental.

Values where we suspect sin(x) is transcendental, but don’t have a formal proof #

There are plenty of "nice" x where we strongly believe sin(x) is transcendental, but haven’t been able to prove it yet (these rely on unproven but widely accepted conjectures like Schanuel’s Conjecture):

• Transcendental numbers in radians: This includes x = e, x = ln(2), x = 1/π. Schanuel’s Conjecture would imply that these sines are transcendental, but no strict proof exists today.
• Irrational numbers in degrees (that don’t reduce to a rational multiple of π radians): For example, x = π degrees (which converts to π²/180 radians) or x = √2 degrees (√2 π/180 radians). Since these x values are transcendental (they involve π multiplied by an algebraic number), we can’t apply Lindemann-Weierstrass directly, but Schanuel’s Conjecture suggests their sines are transcendental.

Quick recap of your initial intuition #

Your gut feeling was mostly right:

• sin(q) (q non-zero rational, radians) is always transcendental.
• For x that’s irrational in radians and not of the form qπ (q rational), if x is algebraic, we can prove sin(x) is transcendental. If x is transcendental, we suspect it’s transcendental but can’t confirm yet.

备注：内容来源于stack exchange，提问作者Gabriel Brown