Mathematics : Trigonometric Functions

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If x and y are acute angles such that cos(x) + cos(y) = sin(x) + sin(y) = $\frac{3}{4}$, then sin(x + y) equals to

• $\frac{2}{5}$

• $\frac{3}{4}$

• $\frac{3}{5}$

• $\frac{4}{5}$

$\mathrm{The} \mathrm{sum} \mathrm{of} \mathrm{the} \mathrm{solutions} \mathrm{in} \left(0, 2\mathrm{\pi }\right) \mathrm{the} \mathrm{equation} \cos \left(\mathrm{x}\right)\cos \left(\frac{\mathrm{\pi }}{3} - \mathrm{x}\right)\cos \left(\frac{\mathrm{\pi }}{3} + \mathrm{x}\right) = \frac{1}{4} \mathrm{is}$

• $4\mathrm{\pi }$

• $\mathrm{\pi }$

• $2\mathrm{\pi }$

• $3\mathrm{\pi }$

$\mathrm{In} \mathrm{any} ∆\mathrm{ABC}, \frac{\left(\mathrm{a} + \mathrm{b} +\hspace{0.17em}\mathrm{c}\right)\left(\mathrm{b} + \mathrm{c} - \mathrm{a}\right)\left(\mathrm{c} +\hspace{0.17em}\mathrm{a} - \mathrm{b}\right)\left(\mathrm{a} +\hspace{0.17em}\mathrm{b} - \mathrm{c}\right)}{4{\mathrm{b}}^2{\mathrm{c}}^2} = ?$

• ${\sin }^2\left(\mathrm{B}\right)$

• ${\cos }^2\left(\mathrm{A}\right)$

• ${\cos }^2\left(\mathrm{B}\right)$

• ${\sin }^2\left(\mathrm{A}\right)$

$\sum _{\mathrm{k} = 1}^6 \left[\sin \left(\frac{{\displaystyle 2\mathrm{k\pi }}}{{\displaystyle 7}} - \mathrm{icos}\left(\frac{{\displaystyle 2\mathrm{k\pi }}}{{\displaystyle 7}}\right)\right)\right] = ?$

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•  - i

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If s$\sin \left(\mathrm{\theta }\right) + \cos \left(\mathrm{\theta }\right) = \mathrm{p}$ and $\tan \left(\mathrm{\theta }\right) + \cot \left(\mathrm{\theta }\right) = \mathrm{q}$, then q(p² - 1) is equal to

• $\frac{1}{2}$

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• 1

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$\tan \left(\frac{\mathrm{\pi }}{5}\right) + 2\tan \left(\frac{2\mathrm{\pi }}{5}\right) + 4\cot \left(\frac{4\mathrm{\pi }}{5}\right)$ is equal to

• $\cot \left(\frac{\mathrm{\pi }}{5}\right)$

• $\cot \left(\frac{2\mathrm{\pi }}{5}\right)$

• $\cot \left(\frac{3\mathrm{\pi }}{5}\right)$

• $\cot \left(\frac{4\mathrm{\pi }}{5}\right)$

If sin(A) + sin(B) + sin(C) = 0 and cos(A) + cos(B) + cos(C) = 0, then cos (A + B) + cos (B + C) + cos(C + A) is equal to

• cos(A + B + C)

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If $\tan \left(\mathrm{\theta }\right) . \tan \left(120° - \mathrm{\theta }\right)\tan \left(120° + \mathrm{\theta }\right) = \frac{1}{\sqrt{3}}$, then $\mathrm{\theta }$ is equal to

• $\frac{\mathrm{n\pi }}{3} + \frac{\mathrm{\pi }}{18}, \mathrm{n} \in \mathrm{Z}$

• $\frac{\mathrm{n\pi }}{3} + \frac{\mathrm{n\pi }}{12}, \mathrm{n} \in \mathrm{Z}$

• $\frac{\mathrm{n\pi }}{12} + \frac{\mathrm{\pi }}{12}, \mathrm{n} \in \mathrm{Z}$

• $\frac{\mathrm{n\pi }}{3} + \frac{\mathrm{\pi }}{6}, \mathrm{n} \in \mathrm{Z}$

${\left(\frac{1 + \cos \left({\displaystyle \frac{\mathrm{\pi }}{8}}\right) - \mathrm{isin}\left(\frac{\mathrm{\pi }}{8}\right)}{1 + \cos \left(\frac{\mathrm{\pi }}{8}\right) + \mathrm{isin}\left(\frac{\mathrm{\pi }}{8}\right)}\right)}^8 = ?$

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• $\frac{1}{2}$