query: In an acute angled triangle, prove that $\sin A + \sin B + \sin C > 2$, where $0<A<B<\frac{\pi}{2}$
positive_passages.docid: 27513
positive_passages.text: Trigonometric bounds
negative_passages.docid: 27954
negative_passages.text: Prove triangle inequality using the properties of absolute value
negative_passages.docid: 120444
negative_passages.text: Trigonometric Triangle Equality
negative_passages.docid: 85944
negative_passages.text: Prove $ \sin x + \frac{ \sin3x }{3} + ... + \frac{ \sin((2n-1)x) }{2n-1} >0 $
negative_passages.docid: 70765
negative_passages.text: If $\frac{\tan(a+b-c)}{\tan(a-b+c)}=\frac{\tan c}{\tan b}$, then $\sin(b-c)=0$ or $\sin 2a + \sin2b + \sin 2c=0$
negative_passages.docid: 137376
negative_passages.text: Proving a trig identity: $\frac{\sin(A + B)}{\sin(A - B)}=\frac{\tan A + \tan B}{\tan A - \tan B}$
negative_passages.docid: 90902
negative_passages.text: Prove $\frac{a+b+c}{abc} \leq \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}$.
negative_passages.docid: 17264
negative_passages.text: Given $a,b,c$ are the sides of a triangle. Prove that $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}<2$
negative_passages.docid: 147684
negative_passages.text: Prove that $\sin \theta=\frac{3 \sin \alpha+\sin^3 \alpha}{1+3\sin^2 \alpha}$ using given condition
negative_passages.docid: 156596
negative_passages.text: Prove $r^2+r_1^2+r_2^2+r_3^2=16R^2-a^2-b^2-c^2$ for a triangle with sides $a$, $b$, $c$, circumradius $R$, inradius $r$, exradii $r_1$, $r_2$, $r_3$
negative_passages.docid: 11711
negative_passages.text: Prove trigonometry identity for $\cos A+\cos B+\cos C$
negative_passages.docid: 143857
negative_passages.text: How to prove absolute summability of sinc function?
negative_passages.docid: 81102
negative_passages.text: Cosine of the sum of two solutions of trigonometric equation $a\cos \theta + b\sin \theta = c$
negative_passages.docid: 32963
negative_passages.text: proving the inequality $\triangle\leq \frac{1}{4}\sqrt{(a+b+c)\cdot abc}$
negative_passages.docid: 33905
negative_passages.text: Prove the following trigonometric identity: $\frac{\tan A+\sec A-1}{\tan A-\sec A+1}=\frac{1+\sin A}{\cos A}$
negative_passages.docid: 123136
negative_passages.text: Proof of the identity: $c\sin \frac{A-B}{2} \equiv (a-b) \cos \frac{C}{2}$
negative_passages.docid: 98745
negative_passages.text: To prove $(\sin\theta + \csc\theta)^2 + (\cos\theta +\sec\theta)^2 \ge 9$