Let $a \in \left( \frac { \pi } { 6 } , \frac { \pi } { 4 } \right)$. Given that
$$\begin{aligned} & x = \tan a \\ & y = \tan ( 2 a ) \\ & z = \tan ( 3 a ) \end{aligned}$$
Which of the following is the correct ordering of these numbers?
A) $x < y < z$
B) $x < z < y$
C) $y < x < z$
D) $z < x < y$
E) $z < y < x$

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 $0 < a < \dfrac{\pi}{2}$ and
$$\cos^{2} a - \cos(2a) = \sin(2a)$$
For the value of $a$ satisfying this equality, which of the following is correct?
A) $\tan a = \dfrac{1}{5}$ B) $\cot a = \dfrac{2}{\sqrt{5}}$ C) $\cos a = \dfrac{1}{\sqrt{5}}$ D) $\operatorname{cosec} a = \sqrt{5}$ E) $\sin(2a) = \dfrac{3}{5}$

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 $x, y$ and $z$ be distinct elements of the set $\left\{\frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}\right\}$ such that
$$\sin x < \tan y < \sec z$$
Which of the following is the correct ordering of $x$, $y$ and $z$?
A) $x < y < z$ B) $y < x < z$ C) $y < z < x$ D) $z < x < y$ E) $z < y < x$