Mathematics : Trigonometric Functions

The smallest value of $5\cos \left(\mathrm{\theta }\right) + 12$ is

• 5

• 12

• 7

• 17

Show that

$\frac{\sin \left(\mathrm{\theta }\right)}{\cos \left(3\mathrm{\theta }\right)} + \frac{{\displaystyle \sin \left(3\mathrm{\theta }\right)}}{{\displaystyle \cos \left(9\mathrm{\theta }\right)}} + \frac{{\displaystyle \sin \left(9\mathrm{\theta }\right)}}{{\displaystyle \cos \left(27\mathrm{\theta }\right)}} = \frac{1}{2}\left(\left(\tan 27\mathrm{\theta }\right) - \tan \left(\mathrm{\theta }\right)\right)$

The equation $\sqrt{3}\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right) = 4$ has

• infinitely many solutions

• no solution

• two solutions

• only one solution

The value of

$\tan \left(\mathrm{\alpha }\right) + 2\tan \left(2\mathrm{\alpha }\right) + 4\tan \left(4\mathrm{\alpha }\right) + ... + 2^{\mathrm{n} - 1}\tan \left(2^{\mathrm{n} - 1}\mathrm{\alpha }\right) + 2^{\mathrm{n}}\cot \left(2^{\mathrm{n}}\mathrm{\alpha }\right)$ is

• $\cot \left(2^{\mathrm{n}}\mathrm{\alpha }\right)$

• $2^{\mathrm{n}}\tan \left(2^{\mathrm{n}}\mathrm{\alpha }\right)$

• 0

• $\cot \left(\mathrm{\alpha }\right)$

If $\tan \left(\frac{\mathrm{\alpha \pi }}{4}\right) = \cot \left(\frac{\mathrm{\beta \pi }}{4}\right)$, then

• $\mathrm{\alpha } + \mathrm{\beta } = 0$

• $\mathrm{\alpha } + \mathrm{\beta } = 2\mathrm{n}$

• $\mathrm{\alpha } + \mathrm{\beta } = 2\mathrm{n} + 1$

• $\mathrm{\alpha } + \mathrm{\beta } = 2(2\mathrm{n} + 1), \forall \mathrm{n} \mathrm{is} \mathrm{an} \mathrm{integer}$

The principal value of ${\sin }^{-1}\left(\tan \left(- \frac{5\mathrm{\pi }}{4}\right)\right)$ is

• $\frac{\mathrm{\pi }}{4}$

• - $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $- \frac{\mathrm{\pi }}{2}$

The value of $\cos \left(\frac{\mathrm{\pi }}{15}\right)\cos \left(\frac{2\mathrm{\pi }}{15}\right)\cos \left(\frac{4\mathrm{\pi }}{15}\right)\cos \left(\frac{8\mathrm{\pi }}{15}\right)$

• $\frac{1}{16}$

• $- \frac{1}{16}$

• 1

• 0

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

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

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

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

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