Which one of the following is a necessary and sufficient condition for
$$\sum _ { k = 1 } ^ { n } \sin \left( \frac { k \pi } { 3 } \right) = \frac { \sqrt { 3 } } { 2 }$$
to be true?
A $n = 1$
B $n$ is a multiple of 3
C $n$ is a multiple of 6
D $n$ is 1 more than a multiple of 3
E $n$ is 1 more than a multiple of 6 F $n$ is 1 more than a multiple of 6 or $n$ is 2 more than a multiple of 6

The equation
$$\sin ^ { 2 } \left( 4 ^ { \cos \theta } \times 60 ^ { \circ } \right) = \frac { 3 } { 4 }$$
has exactly three solutions in the range $0 ^ { \circ } \leq \theta \leq x ^ { \circ }$ What is the range of all possible values of $x$ ?
A $90 \leq x < 120$ B $90 \leq x < 270$ C $120 \leq x < 240$ D $270 \leq x < 300$ E $\quad 300 \leq x < 360$ F $\quad 450 \leq x < 630$

The angle $\theta$ can take any of the values $1 ^ { \circ } , 2 ^ { \circ } , 3 ^ { \circ } , \ldots , 359 ^ { \circ } , 360 ^ { \circ }$. For how many of these values of $\theta$ is it true that
$$\sin \theta \sqrt { 1 + \sin \theta } \sqrt { 1 - \sin \theta } + \cos \theta \sqrt { 1 + \cos \theta } \sqrt { 1 - \cos \theta } = 0$$
A 0
B 1
C 2
D 4
E 93 F 182 G 271 H 360

How many real solutions are there to the equation
$$2 \cos ^ { 4 } \theta - 5 \cos ^ { 2 } \theta + 3 = 0$$
in the interval $0 \leq \theta \leq 2 \pi$ ?

How many solutions are there to
$$( 1 + 3 \cos 3 \theta ) ^ { 2 } = 4$$
in the interval $0 ^ { \circ } \leq \theta \leq 180 ^ { \circ }$ ?

How many solutions are there to the equation
$$\frac { 2 ^ { \tan ^ { 2 } x } } { 4 ^ { \sin ^ { 2 } x } } = 1$$
in the range $0 \leq x \leq 2 \pi$ ?

In this question, $p$ is a real constant. The equation $\sin x \cos ^ { 2 } x = p ^ { 2 } \sin x$ has $n$ distinct solutions in the range $0 \leq x \leq 2 \pi$ Which of the following statements is/are true?
I $n = 3$ is sufficient for $p > 1$ II $n = 7$ only if $- 1 < p < 1$
A none of them B I only C II only D I and II

Given that $0 < \mathrm { x } < \pi$,
$$\sin ^ { 4 } x = \cos ^ { 4 } x$$
What is the sum of the $\mathbf { x }$ values that satisfy this equality?
A) $\frac { 3 \pi } { 2 }$
B) $\frac { 4 \pi } { 3 }$
C) $\frac { 5 \pi } { 4 }$
D) $\pi$
E) $2 \pi$

For $0 \leq \mathrm { x } \leq \pi$,
$$\frac { \sin x \cdot \tan x } { 3 } = 1 - \cos x$$
What is the sum of the $\mathbf { x }$ values that satisfy this equation?
A) $\frac { \pi } { 3 }$
B) $\frac { 2 \pi } { 3 }$
C) $\frac { 4 \pi } { 3 }$
D) $\pi$
E) $2 \pi$

Given that $x \in [ 0, 2 \pi )$, $$\cos ( 5 x ) = \cos ( 3 x ) \cdot \cos ( 2 x )$$ How many different solutions does the equation have?\ A) 3\ B) 6\ C) 8\ D) 11\ E) 12

Where $0 < x < \frac{\pi}{2}$,
$$\frac{1 + \tan x}{\cot x} \cdot \frac{\sin x - \cos x}{\sin x} = 2$$
if this holds, what is the value of $\sin x$?
A) $\frac{1}{3}$
B) $\frac{3}{5}$
C) $\frac{\sqrt{2}}{2}$
D) $\frac{\sqrt{3}}{2}$
E) $\frac{\sqrt{5}}{3}$

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) $\frac { 20 \pi } { 9 }$

Let $\pi < x < 2\pi$,
$$\frac{2\cos^{2} x + 2\sin x}{5\sin(2x)} = \tan x$$
What is the sum of the real numbers $x$ that satisfy this equation?
A) $2\pi$ B) $3\pi$ C) $4\pi$ D) $5\pi$ E) $6\pi$

Let $0 \leq x \leq \pi$ and
$$\sqrt{2}\sin(4x) - \cos(8x) = 1$$
What is the sum of the $x$ values satisfying this equality?
A) $\dfrac{\pi}{3}$ B) $\dfrac{3\pi}{4}$ C) $\pi$ D) $\dfrac{3\pi}{2}$ E) $2\pi$

Let $0 < x < \frac{\pi}{2}$. Given that
$$2 \cdot \cos^{2} x + 9 \cdot \sin^{2} x + 2 \cdot \sin(2x) = 9$$
what is the value of $\cot x$?
A) $\frac{4}{7}$ B) $\frac{7}{6}$ C) $\frac{3}{5}$ D) $\frac{2}{3}$ E) $\frac{5}{2}$