Trigonometric Functions

The principal amplitude of ${\left(\sin \left(40°\right) + \mathrm{icos}\left(40°\right)\right)}^5$

• 70$°$

• - 110$°$

• 110°

• - 70° 

Find the general solution of $\sec \left(\mathrm{\theta }\right) + 1 = \left(2 + \sqrt{3}\right)\tan \left(\mathrm{\theta }\right)$

The real part of ${\left(1 - \cos \left(\mathrm{\theta }\right) + 2\mathrm{isin}\left(\mathrm{\theta }\right)\right)}^{- 1}$

• $\frac{1}{3 + 5\cos \left(\mathrm{\theta }\right)}$

• $\frac{1}{5 - 3\cos \left(\mathrm{\theta }\right)}$

• $\frac{1}{3 - 5\cos \left(\mathrm{\theta }\right)}$

• $\frac{1}{5 + 3\cos \left(\mathrm{\theta }\right)}$

The value of $\sin \left(36°\right)\sin \left(72°\right)\sin \left(108°\right)\sin \left(144°\right)$ is equal to

• $\frac{1}{4}$

• $\frac{1}{16}$

• $\frac{3}{4}$

• $\frac{5}{16}$

If $\sin \left(\mathrm{A}\right) = \frac{1}{\sqrt{10}} \mathrm{and} \sin \left(\mathrm{B}\right) = \frac{1}{\sqrt{5}}$ where A and B are positive acute angles, then A + B is equal to 

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

The expression ${\tan }^2\left(\mathrm{\alpha }\right) + {\cot }^2\left(\mathrm{\alpha }\right)$ is

• $\ge 2$

• $\le 2$

• $\ge - 2$

• None of these

The equation $3{\sin }^2\left(\mathrm{x}\right) + 10\cos \left(\mathrm{x}\right) - 6$ is satisfied, if

• $\mathrm{x} = \mathrm{n\pi } \pm {\cos }^{-1}\left(\frac{1}{3}\right)$

• $\mathrm{x} = 2\mathrm{n\pi } \pm {\cos }^{-1}\left(\frac{1}{3}\right)$

• $\mathrm{x} = \mathrm{n\pi } \pm {\cos }^{-1}\left(\frac{1}{6}\right)$

• $\mathrm{x} = 2\mathrm{n\pi } \pm {\cos }^{-1}\left(\frac{1}{6}\right)$

If ${\mathrm{e}}^{\cos \left(\mathrm{x}\right)} - {\mathrm{e}}^{- \cos \left(\mathrm{x}\right)} = 4$, then the value of $\cos \left(\mathrm{x}\right)$ is

• $\log \left(2 + \sqrt{5}\right)$

• $- \log \left(2 + \sqrt{5}\right)$

• $\log \left(- 2 + \sqrt{5}\right)$

• None of these

If ${\tan }^{-1}\left(\frac{\mathrm{a}}{\mathrm{x}}\right) + {\tan }^{-1}\left(\frac{\mathrm{b}}{\mathrm{x}}\right) = \frac{\mathrm{\pi }}{2}$, then x is equal to

• $\sqrt{\mathrm{ab}}$

• $\sqrt{2\mathrm{ab}}$

• 2ab

• ab

If $\cos \left(\mathrm{\theta } - \mathrm{\alpha }\right) = \mathrm{a}, \cos \left(\mathrm{\theta } - \mathrm{\beta }\right) = \mathrm{b}$, then

${\sin }^2\left(\mathrm{\alpha } - \mathrm{\beta }\right) + 2\mathrm{abcos}\left(\mathrm{\alpha } - \mathrm{\beta }\right)$ is equal to

• a² + b²

• a² - b²

• b² - a²

• - a² - b²