query: Show that $\sin x$ lies between $x-x^3/6$ and $x \;$ $\forall x \in R$
positive_passages.docid: 89265
positive_passages.text: Proving that $x - \frac{x^3}{3!} 0$
negative_passages.docid: 51979
negative_passages.text: Is there a geometrical method to prove $x<\frac{\sin x +\tan x}{2}$?
negative_passages.docid: 118884
negative_passages.text: The sine inequality $\frac2\pi x \le \sin x \le x$ for $0<x<\frac\pi2$
negative_passages.docid: 85249
negative_passages.text: Evaluating: $\lim\limits_{x\to0}\left(\frac{\sin x}{x}\right)^{{6}/{x^{2}}}$
negative_passages.docid: 149102
negative_passages.text: Evaluating $\lim_{x \to 0} x\sin\left(\frac{1}{x}\right)$ or showing it does not exist
negative_passages.docid: 129443
negative_passages.text: $f(\theta) = \cos \theta - \sqrt{3}\sin \theta$ in the form $r\sin(\theta-\alpha)$
negative_passages.docid: 150734
negative_passages.text: Solving $\lim\limits_{x\to0} \frac{x - \sin(x)}{x^2}$ without L'Hospital's Rule.
negative_passages.docid: 150819
negative_passages.text: How to prove by induction that $|\sin(nx)| \leq n|\sin x|$?
negative_passages.docid: 118687
negative_passages.text: Prove an inequality with a $\sin$ function: $\sin(x) > \frac2\pi x$ for $0<x<\frac\pi2$
negative_passages.docid: 70742
negative_passages.text: Evaluate $\lim_{x\rightarrow 0} \frac{\sin (6x)}{\sin(2x)}$ without L'Hopital
negative_passages.docid: 105487
negative_passages.text: Prove that $\sin x \cdot \sin (2x) \cdot \sin(3x) < \tfrac{9}{16}$ for all $x$
negative_passages.docid: 19056
negative_passages.text: Find the limit of $(2\sin x-\sin 2x)/(x-\sin x)$ as $x\to 0$ without L'Hôpital's rule
negative_passages.docid: 112898
negative_passages.text: Derivative of sin(x)/x at $0$ by definition of derivative
negative_passages.docid: 28161
negative_passages.text: Solve $\lim_{x\to 0} \frac{\sin x-x}{x^3}$
negative_passages.docid: 104841
negative_passages.text: Can $\sin n$ get arbitrarily close to $1$ for $n\in\mathbb{N}?$
negative_passages.docid: 53857
negative_passages.text: Showing $\sup \{ \sin n \mid n\in \mathbb N \} =1$
negative_passages.docid: 95673
negative_passages.text: General formula for $\sin\left(k\arcsin (x)\right)$