Prove trigonometric inequality $\sin x\leq 1-\left(\frac{2x}{\pi}-1\right)^2$

Solution 1:

Note that $ f(t)=\frac\pi{\sqrt2}\sin \frac t2,\> t\in [0,\pi/2] $ is a concave function with $f(0)=0$ and $f(\frac\pi2)= \frac\pi2 $, which implies $f(t)\ge t$, i.e.

$$\frac\pi{\sqrt2}\sin \frac t2- t\ge 0 \implies \sin^2\frac t2 \ge (\frac{\sqrt2 t}\pi)^2 \implies 1-\cos t \ge (\frac{2t}\pi)^2 $$

Substitute $x= \frac\pi2+t, \> x\in [0, \pi]$, to obtain

$$\sin x\leq 1-\left(\dfrac{2x}{\pi}-1\right)^2$$

Solution 2:

Concavity of Sine

For $x,y\in[0,\pi]$, $$ \begin{align} \frac{\sin(x)+\sin(y)}2 &=\sin\left(\frac{x+y}2\right)\cos\left(\frac{x-y}2\right)\\[6pt] &\le\sin\left(\frac{x+y}2\right)\tag1 \end{align} $$ Since $\sin(x)$ is continuous, $(1)$ shows that $\sin(x)$ is concave on $[0,\pi]$.

The Inequality

Note that for $x=0$ and $x=\frac\pi4$, $\sin(x)=\frac{2\sqrt2x}\pi$. Thus, since $\sin(x)$ is concave on $\left[0,\frac\pi4\right]$, we have $$ \sin(x)\ge\frac{2\sqrt2x}\pi\tag2 $$ for $x\in\left[0,\frac\pi4\right]$. Thus, for $\frac x2\in\left[0,\frac\pi4\right]$, that is, $x\in\left[0,\frac\pi2\right]$, $$ \begin{align} \cos(x) &=1-2\sin^2\left(\frac x2\right)\tag3\\ &\le1-\frac{4x^2}{\pi^2}\tag4 \end{align} $$ where step $(4)$ is simply an application of $(2)$. Since $(4)$ is even, it is true for $x\in\left[-\frac\pi2,\frac\pi2\right]$. Therefore, for $\frac\pi2-x\in\left[-\frac\pi2,\frac\pi2\right]$, that is, $x\in[0,\pi]$, $$ \begin{align} \sin(x) &=\cos\left(\frac\pi2-x\right)\tag5\\ &\le1-\frac{4\left(\frac\pi2-x\right)^2}{\pi^2}\tag6\\ &=1-\left(\frac{2x}\pi-1\right)^2\tag7 \end{align} $$ where $(6)$ is an application of $(4)$.

Solution 3:

We need to prove that $$2\cos^2\left(\frac{\pi}{4}-\frac{x}{2}\right)\geq\left(\frac{2x}{\pi}-1\right)^2$$ or $$\left(\sqrt2\cos\left(\frac{\pi}{4}-\frac{x}{2}\right)-\frac{2x}{\pi}+1\right)\left(\sqrt2\cos\left(\frac{\pi}{4}-\frac{x}{2}\right)+\frac{2x}{\pi}-1\right)\geq0.$$ Now, show that $$ \sqrt2\cos\left(\frac{\pi}{4}-\frac{x}{2}\right)-\frac{2x}{\pi}+1\geq0$$ and $$\sqrt2\cos\left(\frac{\pi}{4}-\frac{x}{2}\right)+\frac{2x}{\pi}-1\geq0,$$ which we can prove by using one derivative only:

Let $f(x)=\sqrt2\cos\left(\frac{\pi}{4}-\frac{x}{2}\right)-\frac{2x}{\pi}+1.$

Thus, $$f'(x)=\frac{1}{\sqrt2}\cos\left(\frac{x}{2}+\frac{\pi}{4}\right)-\frac{2}{\pi}<0,$$ which says $$f(x)>f(\pi)=0.$$ Let $g(x)=\sqrt2\cos\left(\frac{\pi}{4}-\frac{x}{2}\right)+\frac{2x}{\pi}-1.$

Thus, $$g'(x)=\frac{1}{\sqrt2}\cos\left(\frac{x}{2}+\frac{\pi}{4}\right)+\frac{2}{\pi}>0,$$ which gives $$g(x)\geq g(0)=0$$ and we are done!