query: Find the value of $\sin\frac{\pi}{7}\sin\frac{2\pi}{7}\sin\frac{3\pi}{7}\sin\frac{\pi}{14}\sin\frac{3\pi}{14}\sin\frac{5\pi}{14}$
positive_passages.docid: 18285
positive_passages.text: Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$
negative_passages.docid: 36951
negative_passages.text: How to evaluate the series $1+\frac34+\frac{3\cdot5}{4\cdot8}+\frac{3\cdot5\cdot7}{4\cdot8\cdot12}+\cdots$
negative_passages.docid: 144811
negative_passages.text: Calculate $\int_0^{2\pi} \frac{1}{2+\sin(x)} \ dx$
negative_passages.docid: 120953
negative_passages.text: Show that, $2\arctan\left(\frac{1}{3}\right)+2\arcsin\left(\frac{1}{5\sqrt2}\right)-\arctan\left(\frac{1}{7}\right)=\frac{\pi}{4}$
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: 139327
negative_passages.text: Find the limit of a series of fractions starting with $\frac{1}{2}, \, \frac{1/2}{3/4}, \, \frac{\frac{1}{2}/\frac{3}{4}}{\frac{5}{6}/\frac{7}{8}}$
negative_passages.docid: 99328
negative_passages.text: Value of $\sin (2^\circ)\cdot \sin (4^\circ)\cdot \sin (6^\circ)\cdots \sin (90^\circ) $
negative_passages.docid: 116649
negative_passages.text: Value of: $ \frac {1} {\sqrt {4} + \sqrt {5}} + \frac {1} {\sqrt {5} + \sqrt {6}} + ... + \frac {1} {\sqrt {80} + \sqrt {81}}$
negative_passages.docid: 149866
negative_passages.text: Evaluating $\int_0^{2\pi} \frac {\cos(\theta)}{5-3\cos(\theta)} \, d\theta$ using residue theorem
negative_passages.docid: 60709
negative_passages.text: How do I get $\cos{\theta} \lt \frac{\sin{\theta}}{\theta} \lt 1$?
negative_passages.docid: 60491
negative_passages.text: Evaluate $ \int_{25\pi/4}^{53\pi/4}\frac{1}{(1+2^{\sin x})(1+2^{\cos x})}dx $
negative_passages.docid: 59791
negative_passages.text: Determine the exact value of equations involving more two trig variables
negative_passages.docid: 47706
negative_passages.text: Find the value of $ \sum_{n=1}^{\infty} \frac{n^3}{3^n} $
negative_passages.docid: 22505
negative_passages.text: Evaluate : $1+\frac{2}{6}+\frac{2\cdot 5}{6\cdot 12}+\frac{2\cdot 5\cdot 8}{6\cdot12\cdot 18}+\ldots$
negative_passages.docid: 115759
negative_passages.text: Evaluating this integral : $ \int \frac {1-7\cos^2x} {\sin^7x \cos^2x} dx $
negative_passages.docid: 125171
negative_passages.text: Help with this trigonometry problem?
negative_passages.docid: 140342
negative_passages.text: $\frac {1}{5}-\frac{1.4}{5.10}+\frac{1.4.7}{5.10.15}-\cdot\cdot\cdot\cdot\cdot\cdot\cdot$