query: How to show $ \sin x \geq \frac{2x}{\pi}, x \in [0, \frac{\pi}{2}]$?
positive_passages.docid: 11387
positive_passages.text: Mean Value Theorem: $\frac{2}{\pi}<\frac{\sin x}{x}<1$
negative_passages.docid: 29322
negative_passages.text: Evaluate $\int_{0}^{\frac{\pi}{2}} \frac{\sin^2 nx}{\sin x} \text{d}x$
negative_passages.docid: 73771
negative_passages.text: An inequality for a sum of sine functions
negative_passages.docid: 36366
negative_passages.text: Prove $\int_0^\infty \frac{\sin^4x}{x^4}dx = \frac{\pi}{3}$
negative_passages.docid: 49054
negative_passages.text: Show that $\left | \sin a - \sin b \right | \leq \left | a-b \right |$
negative_passages.docid: 60559
negative_passages.text: How prove this $x^3<\sin^2{x}\tan{x},x\in\left(0,\dfrac{\pi}{2}\right)$
negative_passages.docid: 142114
negative_passages.text: Limit as $x\to 2$ of $\frac{\cos(\frac \pi x)}{x-2} $
negative_passages.docid: 129892
negative_passages.text: show $\int_0^{\infty} \frac{x\sin ax}{x^2+t^2}dx = \frac{\pi}{2}e^{-at}$
negative_passages.docid: 35529
negative_passages.text: Prove that $\sin x+2x\ge\frac{3x(x+1)}{\pi}$ for all $x\in [0,\frac{\pi}{2}]$
negative_passages.docid: 150819
negative_passages.text: How to prove by induction that $|\sin(nx)| \leq n|\sin x|$?
negative_passages.docid: 158279
negative_passages.text: $\int_0^{\pi/2}\ln(\sin(x))$?
negative_passages.docid: 106544
negative_passages.text: Solve $ \frac{2}{\pi}\int_{-\pi}^\pi\frac{\sin\frac{9x}{2}}{\sin\frac{x}{2}}dx $
negative_passages.docid: 17769
negative_passages.text: Show that $\frac{\sin x}{\cos 3x}+\frac{\sin 3x}{\cos 9x}+\frac{\sin 9x}{\cos 27x} = \frac{1}{2}\left(\tan 27x-\tan x\right)$
negative_passages.docid: 152640
negative_passages.text: show that $\lim(\pi(x)/x) = 0$
negative_passages.docid: 54769
negative_passages.text: Distribution of sine of uniform random variable on $[0, 2\pi]$
negative_passages.docid: 137060
negative_passages.text: Evaluating $\int_0^{\tfrac{\pi}{4}}\ln(\cos x-\sin x)\ln(\cos x) dx - \int_0^{\tfrac{\pi}{4}}\ln(\cos x+\sin x)\ln(\sin x)dx=\frac{G\ln 2}{2}$
negative_passages.docid: 21088
negative_passages.text: Prove $\int\limits_{0}^{\pi/2}\frac{dx}{1+\sin^2{(\tan{x})}}=\frac{\pi}{2\sqrt{2}}\bigl(\frac{e^2+3-2\sqrt{2}}{e^2-3+2\sqrt{2}}\bigr)$