TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-02

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The zero sets of sine and cosine and the least positive common period 2 pi

Statement

For every real $x$ , $\sin x = 0 ⟺ x = m π for some m \in Z, \cos x = 0 ⟺ x = (m + 1 / 2) π for some m \in Z .$ Both sine and cosine have period $2 π$ , and no smaller positive number is a common period.

Facts & Assumptions

Given: A real $x$ .

[L1]

$π$ is the first positive sine zero, and sine is positive on $(0, π)$ (Pi is the first positive zero of sine).

[L2]

Shifts by $π$ negate sine and cosine, while shifts by $π / 2$ exchange them up to sign (Quarter-turn values and shifts by pi/2 and pi).

[L3]

Every real has an integer part; every integer is a natural number or the negative of a natural number; natural induction and the integer-power laws are valid (Integer part: for every real $x$ there is exactly one integer $m$ with $m \leq x < m + 1$ , Integer powers $a^{m}$ , The principle of mathematical induction, Laws of integer exponents).

Proof

technique · direct

1.1L2L3algebra

Natural induction applied to the $π$ shift gives $\sin (x + n π) = (- 1)^{n} \sin x$ and $\cos (x + n π) = (- 1)^{n} \cos x$ for every natural $n$ . Applying these identities at $x - n π$ gives the matching backward shifts; since every integer is $n$ or $- n$ and $(- 1)^{- n} = (- 1)^{n}$ , the displayed identities hold for every integer $m$ .

2.1L1step 1.1L3

Choose an integer $m$ with $m π \leq x < (m + 1) π$ and put $r = x - m π \in [0, π)$ . By [L1] and step 1.1, $\sin x = 0$ exactly when $r = 0$ , hence exactly when $x = m π$ .

3.1step 2.1L2

The quarter-turn shift converts the sine zero set into $\cos x = 0$ exactly when $x = (m + 1 / 2) π$ .

4.1step 2.1step 3.1L1L2

Step 1.1 with $m = 2$ gives period $2 π$ . A positive common period $T$ has $\sin T = 0$ , hence $T = m π$ by step 2.1; $m = 1$ fails for cosine because $\cos π = - 1$ , so $m \geq 2$ .

5.1step 4.1∎

Therefore $2 π$ is the least positive common period.

The zero sets of sine and cosine and the least positive common period 2 pi

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources