For $0 < x < 2 \pi$, what is the sum of all values of $x$ that simultaneously satisfy the equation $4 \cos ^ { 2 } x - 1 = 0$ and the inequality $\sin x \cos x < 0$? [3 points]
(1) $\frac { 10 } { 3 } \pi$
(2) $3 \pi$
(3) $\frac { 8 } { 3 } \pi$
(4) $\frac { 7 } { 3 } \pi$
(5) $2 \pi$

For the function $f(x) = \sin\frac{\pi}{4}x$, find the sum of all natural numbers $x$ satisfying the inequality $$f(2+x)f(2-x) < \frac{1}{4}$$ for $0 < x < 16$. [3 points]

Using the relation $2 ( 1 - \cos x ) < x ^ { 2 } , x ^ { 1 } 0$ or otherwise, prove that $\sin ( \tan x ) > x \forall x \hat { \mathrm { I } } [ 0 , \pi / 4 ]$.

The set of all possible values of $\theta$ in the interval $( 0 , \pi )$ for which the points $( 1,2 )$ and $( \sin \theta , \cos \theta )$ lie on the same side of the line $x + y = 1$ is?
(1) $\left( 0 , \frac { \pi } { 2 } \right)$
(2) $\left( \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } \right)$
(3) $\left( 0 , \frac { 3 \pi } { 4 } \right)$
(4) $\left( 0 , \frac { \pi } { 4 } \right)$

All possible values of $\theta \in [ 0,2 \pi ]$ for which $\sin 2 \theta + \tan 2 \theta > 0$ lie in :
(1) $\left( 0 , \frac { \pi } { 2 } \right) \cup \left( \pi , \frac { 3 \pi } { 2 } \right)$
(2) $\left( 0 , \frac { \pi } { 2 } \right) \cup \left( \frac { \pi } { 2 } , \frac { 3 \pi } { 4 } \right) \cup \left( \pi , \frac { 7 \pi } { 6 } \right)$
(3) $\left( 0 , \frac { \pi } { 4 } \right) \cup \left( \frac { \pi } { 2 } , \frac { 3 \pi } { 4 } \right) \cup \left( \pi , \frac { 5 \pi } { 4 } \right) \cup \left( \frac { 3 \pi } { 2 } , \frac { 7 \pi } { 4 } \right)$
(4) $\left( 0 , \frac { \pi } { 4 } \right) \cup \left( \frac { \pi } { 2 } , \frac { 3 \pi } { 4 } \right) \cup \left( \frac { 3 \pi } { 2 } , \frac { 11 \pi } { 6 } \right)$

Q64. Let $| \cos \theta \cos ( 60 - \theta ) \cos ( 60 + \theta ) | \leq \frac { 1 } { 8 } , \theta \epsilon [ 0,2 \pi ]$. Then, the sum of all $\theta \epsilon [ 0,2 \pi ]$, where $\cos 3 \theta$ attains its maximum value, is :
(1) $15 \pi$
(2) $18 \pi$
(3) $6 \pi$
(4) $9 \pi$

Let $0 \leq \theta \leq 2 \pi$. All $\theta$ satisfying $\sin 2 \theta > \sin \theta$ and $\cos 2 \theta > \cos \theta$ can be expressed as $a \pi < \theta < b \pi$, where $a$ and $b$ are real numbers. What is the value of $b - a$?
(1) $\frac { 1 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 3 } { 4 }$
(5) 1

18. The angle $x$ is measured in radians and is such that $0 \leq x \leq \pi$.
The total length of any intervals for which $- 1 \leq \tan x \leq 1$ and $\sin 2 x \geq 0.5$ is
A $\frac { \pi } { 12 }$
B $\frac { \pi } { 6 }$
C $\frac { \pi } { 4 }$
D $\frac { \pi } { 3 }$
E $\frac { 5 \pi } { 12 }$ F $\frac { \pi } { 2 }$ G $\quad \frac { 5 \pi } { 6 }$

Find the fraction of the interval $0 \leq \theta \leq \pi$ for which the inequality
$$\left(\sin(2\theta) - \frac{1}{2}\right)(\sin\theta - \cos\theta) \geq 0$$
is satisfied.