query: Proof of inequality $|x \sin \alpha + y \cos \alpha| \leq \sqrt{x^2 + y^2}$
positive_passages.docid: 45619
positive_passages.text: Prove: $|a\sin x+b \cos x|\leq \sqrt{a^2+b^2}$
negative_passages.docid: 28218
negative_passages.text: Elementary proof for $\sqrt[3]{xyz} \leq \dfrac{x+y+z}{3}$
negative_passages.docid: 125817
negative_passages.text: show this inequality $(x+y)^3+(y+z)^3+(z+w)^3+(w+x)^3\ge 8(x^2y+y^2z+z^2w+w^2x)$
negative_passages.docid: 81775
negative_passages.text: Prove $a^\alpha b^{1-\alpha} \le \alpha a + (1 - \alpha)b, \; a,b > 0,\; 0 < \alpha < 1$
negative_passages.docid: 73771
negative_passages.text: An inequality for a sum of sine functions
negative_passages.docid: 106979
negative_passages.text: Prove the inequality and limitation
negative_passages.docid: 44233
negative_passages.text: How to prove this inequality $ x + \frac{1}{x} \geq 2 $
negative_passages.docid: 147785
negative_passages.text: Proving inequality $x^xy^y \geq (\frac{x+y}{2})^{x+y}$
negative_passages.docid: 145632
negative_passages.text: Proving $\frac{1}{\cos^2\alpha}<\frac{\tan\beta-\tan\alpha}{\beta-\alpha}<\frac{1}{\cos^2\beta}$
negative_passages.docid: 129890
negative_passages.text: Proving inequality $|\sin(n\theta) |\le n\sin\theta$
negative_passages.docid: 156513
negative_passages.text: Prove that inequality $\sum _{cyc}\frac{a^2+b^2}{a+b}\le \frac{3\left(a^2+b^2+c^2\right)}{a+b+c}$
negative_passages.docid: 25458
negative_passages.text: Prove inequalities $2\sqrt{n+1}-2 \leq 1 + 1/\sqrt{2} + 1/\sqrt{3} + .... + 1/\sqrt{n}\leq 2\sqrt{n}-1$
negative_passages.docid: 72470
negative_passages.text: Prove that $2|ab| \leq a^2 + b^2$ and $|a|+|b| \leq \sqrt {2}(a^2+b^2)^{1/2}$
negative_passages.docid: 49750
negative_passages.text: Proof of the square root inequality $2\sqrt{n+1}-2\sqrt{n}<\frac{1}{\sqrt{n}}<2\sqrt{n}-2\sqrt{n-1}$
negative_passages.docid: 88687
negative_passages.text: Proof of the inequality $||a+b|^p - |a|^p| \leq \epsilon |a|^p + C_{\epsilon} |b|^p$
negative_passages.docid: 66247
negative_passages.text: Proving inequality $\left( \frac {x} {y}\right) ^{x}\left( \frac {1-x} {1-y}\right) ^{1-x}\geqslant 1$
negative_passages.docid: 5369
negative_passages.text: Proving the inequality $|a-b| \leq |a-c| + |c-b|$ for real $a,b,c$