Mathematics : Trigonometric Functions

The number of solutions of $2\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right) = 3$

• 1

• 2

• infinite

• no solution

Let $\tan \left(\mathrm{\alpha }\right) = \frac{\mathrm{a}}{\mathrm{a} + 1} \mathrm{and} \tan \left(\mathrm{\beta }\right) = \frac{1}{2\mathrm{a} + 1}$, then $\mathrm{\alpha } + \mathrm{\beta }$ is

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{3\mathrm{\pi }}{4}$

If $\mathrm{\theta } + \mathrm{\varphi } = \frac{\mathrm{\pi }}{4}, \mathrm{then} \left(1 + \tan \left(\mathrm{\theta }\right)\right) \left(1 + \tan \left(\mathrm{\varphi }\right)\right)$ is equal to

• 1

• 2

• 5/2

• 1/3

If $\sin \left(\mathrm{\theta }\right) \mathrm{and} \cos \left(\mathrm{\theta }\right)$ are the roots of the equation ax² - bx + c = 0, then a, b and c satisfy the relation

• a² + b² + 2ac = 0

• a² - b² + 2ac = 0

• a² + c² + 2ab = 0

• a² - b² - 2ac = 0

If $\sin \left(\mathrm{\theta }\right) + \cos \left(\mathrm{\theta }\right) = 0 \mathrm{and} 0 < \mathrm{\theta } < \mathrm{\pi }, \mathrm{then} \mathrm{\theta }$

• 0

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{3\mathrm{\pi }}{4}$

The value of $\cos \left(15°\right) - \sin \left(15°\right)$ is

• 0

• $\frac{1}{\sqrt{2}}$

• $- \frac{1}{\sqrt{2}}$

• $\frac{1}{2\sqrt{2}}$

The period of the function f(x) = $\cos \left(4\mathrm{x}\right) + \tan \left(3\mathrm{x}\right)$ is

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

If $\mathrm{x} + \frac{1}{\mathrm{x}} = 2\cos \left(\mathrm{\theta }\right)$, then for any integer n, ${\mathrm{x}}^{\mathrm{n}} + \frac{1}{{\mathrm{x}}^{\mathrm{n}}}$ is equal to

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

• $2\sin \left(\mathrm{n\theta }\right)$

• $2\mathrm{icos}\left(\mathrm{n\theta }\right)$

• $2\mathrm{isin}\left(\mathrm{n\theta }\right)$

Prove that the equation cos(2x) + asin(x) = 2a - 7 possesses a solution if $2 \le \mathrm{a} \le 6$.

Find the values of x. $\left(- \mathrm{\pi } < \mathrm{x} < \mathrm{\pi }, \mathrm{x} \ne 0\right)$ satisfying the equation,

${\mathrm{g}}^{1 + \left|\cos \left(\mathrm{x}\right)\right| + \left|{\cos }^2\left(\mathrm{x}\right)\right| + ... \infty = 4^3}$