---
tag: 1JY0FYD2DX
kind: statement
role: proposition
name: A trigonometric inequality
msc: [03F30]
sources:
  - wikipedia: en:18881@r1373903526
    locator: Mathematical induction, section Examples > A trigonometric inequality
    note: adopted
    mention: wikipedia://en/18881#H97X
    sections: [Statement]
inventory: [8RV0MPYF5C]
---

## Statement

Induction is often used to prove inequalities. As an example, we prove that $\left|\sin nx\right| \leq n\left|\sin x\right|$ for any [[real number|08J2M1JYAG]] $x$ and natural number $n$.

At first glance, it may appear that a more general version, $\left|\sin nx\right| \leq n\left|\sin x\right|$ for *any* real numbers $n,x$, could be proven without induction; but the case $n = \frac{1}{2},\, x=\pi$ shows it may be false for non-integer values of $n$. This suggests we examine the statement specifically for *natural* values of $n$, and induction is the readiest tool.