Mathematics : Trigonometric Functions

If f : $\mathrm{R} \to \mathrm{R}$ be the signum function and g : $\mathrm{R} \to \mathrm{R}$ be the greatest integer function, then $\sin \left\{\mathrm{\pi }\left(\left(\mathrm{fog}\right)\left(\frac{1}{2}\right)\right)\right\}$ is equal to

• 1

• $\frac{\sqrt{3}}{2}$

• 0

• $\frac{1}{\sqrt{2}}$

The number of solution of $\tan \left(5\mathrm{\pi cos}\left(\mathrm{\theta }\right)\right) = \cot \left(5\mathrm{\pi sin}\left(\mathrm{\theta }\right)\right) \mathrm{for} \mathrm{\theta } \mathrm{in} \left(0, 2\mathrm{\pi }\right)$ will be

• 28

• 14

• 4

• 2

AB is a vertical pole. The end A is on the ground level. C is the middle point of AB and P is a point on the ground level. The portion BC subtends an angle $\mathrm{\beta }$ at P. If AP = nAB, then $\tan \left(\mathrm{\beta }\right)$ is equal to

• $\frac{\mathrm{n}}{2{\mathrm{n}}^2 + 1}$

• $\frac{\mathrm{n}}{{\mathrm{n}}^2 + 1}$

• $\frac{\mathrm{n}}{\mathrm{n} + 1}$

• None of these

$\tan \left(\mathrm{\theta }\right) + \cot \left(\mathrm{\theta }\right) = 2$ = 2, then $\sin \left(\mathrm{\theta }\right)$ is equal to

• $\frac{1}{\sqrt{2}}$

• $\frac{1}{\sqrt{3}}$

• $\frac{1}{2}$

• 1

$$\begin{gathered} \mathrm{If} \mathrm{\theta } = \frac{\mathrm{\pi }}{6}, \mathrm{then} \mathrm{the} 10\mathrm{th} \mathrm{term} \mathrm{of} 1 + \left(\cos \left(\mathrm{\theta }\right) + \mathrm{isin}\left(\mathrm{\theta }\right)\right) + {\left(\cos \left(\mathrm{\theta }\right) + \mathrm{isin}\left(\mathrm{\theta }\right)\right)}^2 + \\ {\left(\cos \left(\mathrm{\theta }\right) + \mathrm{isin}\left(\mathrm{\theta }\right)\right)}^3 + ...\mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• $\mathrm{i}$

• - 1

• 1

• $- \mathrm{i}$

$\frac{\sin \left(5\mathrm{\theta }\right)}{\sin \left(\mathrm{\theta }\right)} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $16co{s}^4\left(\theta \right) - 12co{s}^2\left(\theta \right)+ 1$

• $16{\cos }^4\left(\mathrm{\theta }\right)+ 12{\cos }^2\left(\mathrm{\theta }\right) + 1$

• $16{\cos }^4\left(\mathrm{\theta }\right) - 12{\cos }^2\left(\mathrm{\theta }\right) - 1$

• $16{\cos }^4\left(\mathrm{\theta }\right) + 12{\cos }^2\left(\mathrm{\theta }\right) - 1$

${\cos }^2\left(\frac{\mathrm{\pi }}{6} + \mathrm{\theta }\right) - {\sin }^2\left(\frac{\mathrm{\pi }}{6} + \mathrm{\theta }\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{1}{2}\cos \left(2\mathrm{\theta }\right)$

• 0

• $- \frac{1}{2}\cos \left(2\mathrm{\theta }\right)$

• $\frac{1}{2}$

$\mathrm{In} ∆\mathrm{ABC}, \frac{\cos \left(\mathrm{C}\right) + \cos \left(\mathrm{A}\right)}{\mathrm{c} + \mathrm{a}} + \frac{\cos \left(\mathrm{B}\right)}{\mathrm{b}} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{1}{\mathrm{a}}$

• $\frac{1}{\mathrm{b}}$

• $\frac{\mathrm{c} + \mathrm{a}}{\mathrm{b}}$

• 1

$\mathrm{In} ∆\mathrm{ABC}, \frac{\mathrm{a}}{{\mathrm{b}}^2 - {\mathrm{c}}^2} + \frac{\mathrm{c}}{{\mathrm{b}}^2 - {\mathrm{a}}^2} = 0, \mathrm{then} \mathrm{B} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{2\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{3}$

$\mathrm{In} ∆\mathrm{ABC},{\mathrm{a}}^2\sin \left(2\mathrm{C}\right) + {\mathrm{c}}^2\sin \left(2\mathrm{A}\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

• $∆$

• $2∆$

• $3∆$

• $4∆$