Mathematics : Trigonometric Functions

$$\begin{gathered} \mathrm{The} \mathrm{set} \mathrm{of} \mathrm{solutions} \mathrm{of} \mathrm{the} \mathrm{system} \mathrm{of} \mathrm{equations} \\ \mathrm{x} + \mathrm{y} = \frac{2\mathrm{\pi }}{3} \\ \mathrm{and} \cos \left(\mathrm{x}\right) + \cos \left(\mathrm{y}\right) = \frac{3}{2}, \\ \mathrm{where} \mathrm{x}, \mathrm{y} \mathrm{are} \mathrm{real}, \mathrm{is} \end{gathered}$$

• $\left\{\left(\mathrm{x}, \mathrm{y}\right)\cos \left(\frac{\mathrm{x} - \mathrm{y}}{2}\right) = \frac{1}{2}\right\}$

• $\left\{\left(\mathrm{x}, \mathrm{y}\right)\sin \left(\frac{\mathrm{x} - \mathrm{y}}{2}\right) = \frac{1}{2}\right\}$

• $\left\{\left(\mathrm{x}, \mathrm{y}\right)\cos \left(\mathrm{x} - \mathrm{y}\right) = \frac{1}{2}\right\}$

• Empty set

If in the angles of a triangle are in the ratio1 : 1 : 4, then the ratio of the perimeter of the triangle to its largest side is

• $\sqrt{2} + 2 : \sqrt{3}$

• 3 : 2

• $\sqrt{3} + 2 . \sqrt{2}$

• $2 + \sqrt{3} : \sqrt{3}$

$\mathrm{If} \cos \left(\mathrm{x}\right) = \tan \left(\mathrm{y}\right), \cot \left(\mathrm{y}\right) = \tan \left(\mathrm{z}\right) \mathrm{and} \cot \left(\mathrm{z}\right) = \tan \left(\mathrm{x}\right) \mathrm{then} \sin \left(\mathrm{x}\right) = ?$

• $\frac{\sqrt{5} + 1}{4}$

• $\frac{\sqrt{5} - 1}{4}$

• $\frac{\sqrt{5} + 1}{2}$

• $\frac{\sqrt{5} - 1}{2}$

$\tan \left(81°\right) - \tan \left(63°\right) - \tan \left(27°\right) + \tan \left(9°\right) = ?$

• 6

• 0

• 2

• 4

If x and y are acute angles such that cos(x) + cos(y) = $\frac{3}{2}$sin(x) + sin(y) = $\frac{3}{4}$, then sin(x + y) equals to

• $\frac{2}{5}$

• $\frac{3}{4}$

• $\frac{3}{5}$

• $\frac{4}{5}$

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

• $4\mathrm{\pi }$

• $\mathrm{\pi }$

• $2\mathrm{\pi }$

• $3\mathrm{\pi }$

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

• ${\sin }^2\left(\mathrm{B}\right)$

• ${\cos }^2\left(\mathrm{A}\right)$

• ${\cos }^2\left(\mathrm{B}\right)$

• ${\sin }^2\left(\mathrm{A}\right)$

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

•  - 1

• 0

•  - i

• i

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

• $\frac{1}{2}$

• 2

• 1

• 3

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

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

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

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

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