Mathematics : Trigonometric Functions

$\mathrm{The} \mathrm{solution} \mathrm{set} \mathrm{of} \left(5 + 4\cos \left(\mathrm{\theta }\right)\right)\left(2\cos \left(\mathrm{\theta }\right) + 1\right) = 0 \mathrm{in} \mathrm{the} \mathrm{interval} \left[0, 2\mathrm{\pi }\right], \mathrm{is} :$

• $\left\{\frac{\mathrm{\pi }}{3}, \frac{2\mathrm{\pi }}{3}\right\}$

• $\left\{\frac{\mathrm{\pi }}{3}, \mathrm{\pi }\right\}$

• $\left\{\frac{2\mathrm{\pi }}{3}, \frac{4\mathrm{\pi }}{3}\right\}$

• $\left\{\frac{2\mathrm{\pi }}{3}, \frac{5\mathrm{\pi }}{3}\right\}$

$\mathrm{If} \frac{\tan \left(3\mathrm{A}\right)}{\tan \left(\mathrm{A}\right)} = \mathrm{a}, \mathrm{then} \frac{\sin \left(3\mathrm{A}\right)}{\sin \left(\mathrm{A}\right)} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{2\mathrm{a}}{\mathrm{a} + 1}$

• $\frac{2\mathrm{a}}{\mathrm{a} - 1}$

• $\frac{\mathrm{a}}{\mathrm{a} + 1}$

• $\frac{\mathrm{a}}{\mathrm{a} - 1}$

If cos(A - B) = 3/5 and tan(A)tan(B) = 2, then which one of the following is true ?

• $\sin \left(\mathrm{A} + \mathrm{B}\right) = \frac{1}{5}$

• $\sin \left(\mathrm{A} + \mathrm{B}\right) = - \frac{1}{5}$

• $\cos \left(\mathrm{A} - \mathrm{B}\right) = \frac{1}{5}$

• $\cos \left(\mathrm{A} + \mathrm{B}\right) = - \frac{1}{5}$

The angle of elevation of an object from a point P on the level ground is $\mathrm{\alpha }$. Moving d metres on the ground towards the object, the angle of elevation is found to be $\mathrm{\beta }$. Then the height (in metres) of the object is

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

• $\mathrm{dcot}\left(\mathrm{\beta }\right)$

• $\frac{\mathrm{d}}{\cot \left(\mathrm{\alpha }\right) + \cot \left(\mathrm{\beta }\right)}$

• $\frac{\mathrm{d}}{\cot \left(\mathrm{\alpha }\right) - \cot \left(\mathrm{\beta }\right)}$

A vertical pole subtends an angle ${\tan }^{-1}\left(\frac{1}{2}\right)$ at a point P on the ground. If the angles substended by the upper half and the lower half of the pole at P are respectively $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ , then $\left(\tan \left(\mathrm{\alpha }\right), \tan \left(\mathrm{\beta }\right)\right)$ is equal to

• $\left(\frac{1}{4}, \frac{1}{5}\right)$

• $\left(\frac{1}{5}, \frac{2}{9}\right)$

• $\left(\frac{2}{9}, \frac{1}{4}\right)$

• $\left(\frac{1}{4}, \frac{2}{9}\right)$