query: Hensen inequality in trigonometry: $\sin A + \sin B + \sin C \leq \frac{3}{2} \cdot \sqrt[2]{3} $
positive_passages.docid: 29424
positive_passages.text: Maximum value of $\sin A+\sin B+\sin C$?
negative_passages.docid: 58969
negative_passages.text: A proof for the inequality $\sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2} $ for all $t \geq 2$
negative_passages.docid: 79491
negative_passages.text: Inequality $(a+b)^2 + (a+b+4c)^2\ge \frac{kabc}{a+b+c}$ for $a,b,c \in\mathbb{R}$
negative_passages.docid: 15116
negative_passages.text: Proof of the inequality $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \geq \frac{3}{2}$
negative_passages.docid: 32963
negative_passages.text: proving the inequality $\triangle\leq \frac{1}{4}\sqrt{(a+b+c)\cdot abc}$
negative_passages.docid: 127759
negative_passages.text: Prove that $\frac{\sin(a)}{\sin(b)} < \frac{a}{b} < \frac{\tan(a)}{\tan(b)}$ where $0 < b < a < \frac{\pi}{2}$
negative_passages.docid: 93599
negative_passages.text: General Proof for the triangle inequality
negative_passages.docid: 156555
negative_passages.text: Inequality $\sqrt{1+x^2}+\sqrt{1+y^2}+\sqrt{1+z^2} \le \sqrt{2}(x+y+z)$
negative_passages.docid: 111903
negative_passages.text: An inequality for sides of a triangle
negative_passages.docid: 135120
negative_passages.text: Oppenheim's Inequality for triangles, American Mathematical Monthly problems
negative_passages.docid: 134535
negative_passages.text: Inequality $\frac{a}{b} + \frac{b}{c} + \frac{c}{a} \geq a + b+ c$ when $abc = 1$
negative_passages.docid: 22237
negative_passages.text: Proof of triangle inequality
negative_passages.docid: 56101
negative_passages.text: Prove the inequality $\frac{a}{c+a-b}+\frac{b}{a+b-c}+\frac{c}{b+c-a}\ge{3}$
negative_passages.docid: 77576
negative_passages.text: Using the triangle inequalities to prove a bound in the complex plane
negative_passages.docid: 140299
negative_passages.text: Inequality based on triangle: $\frac{3}{2}\le\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}<2$
negative_passages.docid: 51979
negative_passages.text: Is there a geometrical method to prove $x<\frac{\sin x +\tan x}{2}$?
negative_passages.docid: 39015
negative_passages.text: Prove the triangle inequality on the sphere $S^2$ in $\mathbb{R}^3$.