114- The sum of the solutions of the trigonometric equation $\sqrt{7}\sin x + \sqrt{3}\cos x = \sqrt{7}$ on the interval $[-\pi, 2\pi]$ is which of the following?
(1) $\dfrac{\pi}{2}$ (2) $\dfrac{7\pi}{3}$ (3) $\dfrac{9\pi}{4}$ (4) $\dfrac{11\pi}{6}$

13. In the trigonometric equation $m(\cos x - \sin x) - 3\sqrt{6}\sin(2x) = \sqrt{6}$, if $\cos(x + \dfrac{\pi}{4}) = \dfrac{1}{\sqrt{3}}$, what is the value of $m$?
\[ \text{(1) } -6 \qquad \text{(2) } -3 \qquad \text{(3) } 6 \qquad \text{(4) } 3 \]

For $0 \leq x < 2 \pi$, the number of solutions of the equation $$\sin ^ { 2 } x + 2 \cos ^ { 2 } x + 3 \sin x \cos x = 0$$ is
(A) 1 .
(B) 2 .
(C) 3 .
(D) 4 .

The number of solutions of the equation $x + 2 \tan x = \frac { \pi } { 2 }$ in the interval $[ 0,2 \pi ]$ is
(1) 3
(2) 4
(3) 2
(4) 5

If $n$ is the number of solutions of the equation $2 \cos x \left(4 \sin \frac { \pi } { 4 } + x \sin \frac { \pi } { 4 } - x\right) - 1 = 1 , x \in [0 , \pi]$ and $S$ is the sum of all these solutions, then the ordered pair $(n , S)$ is :
(1) $\left(2 , \frac { 8 \pi } { 9 }\right)$
(2) $\left(3 , \frac { 13 \pi } { 9 }\right)$
(3) $\left(2 , \frac { 2 \pi } { 3 }\right)$
(4) $\left(3 , \frac { 5 \pi } { 3 }\right)$

If $2 \sin ^ { 3 } x + \sin 2 x \cos x + 4 \sin x - 4 = 0$ has exactly 3 solutions in the interval $\left[ 0 , \frac { \mathrm { n } \pi } { 2 } \right] , \mathrm { n } \in \mathrm { N }$, then the roots of the equation $x ^ { 2 } + n x + ( n - 3 ) = 0$ belong to :
(1) $( 0 , \infty )$
(2) $( - \infty , 0 )$
(3) $\left( - \frac { \sqrt { 17 } } { 2 } , \frac { \sqrt { 17 } } { 2 } \right)$
(4) $Z$

How many real numbers $x$ satisfy $\sin\left(x + \frac{\pi}{6}\right) = \sin x + \sin\frac{\pi}{6}$ and $0 \leq x < 2\pi$?
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5 or more

This question uses radians. Find the number of distinct values of $x$ that satisfy the equation
$$( x + 1 ) ( 3 - x ) = 2 ( 1 - \cos ( \pi x ) )$$
A 2 B 3 C 4 D 5 E 6 F 7

For $0 \leq x \leq 2 \pi$, $\cos \mathrm { x } + \sin 2 \mathrm { x } = \cot \mathrm { x }$ What is the sum of the $\mathbf { x }$ values that satisfy this equation?
A) $2 \pi$
B) $3 \pi$
C) $4 \pi$
D) $\frac { 5 \pi } { 2 }$
E) $\frac { 7 \pi } { 2 }$

For $0 < x < \pi$,
$$\frac { \sin x \cdot \cos x } { \sin x + \cos x } = \frac { \sin x - \cos x } { 2 }$$
What is the sum of the $\mathbf { x }$ values that satisfy the equality?
A) $\frac { \pi } { 2 }$ B) $\frac { 5 \pi } { 4 }$ C) $\frac { 7 \pi } { 4 }$ D) $\pi$ E) $2 \pi$

Let $0 \leq x \leq \frac { 3 \pi } { 2 }$. Given that
$$| \sin x | = \cos \left( 50 ^ { \circ } \right)$$
what is the sum of the $x$ values that satisfy this equation?
A) $\frac { 13 \pi } { 18 }$
B) $\frac { 11 \pi } { 90 }$
C) $\frac { 3 \pi } { 2 }$
D) $\frac { 31 \pi } { 18 }$
E) $\frac { 20 \pi } { 9 }$