If $0 < x , y < \pi$ and $\cos x + \cos y - \cos ( x + y ) = \frac { 3 } { 2 }$, then $\sin x + \cos y$ is equal to:
(1) $\frac { 1 } { 2 }$
(2) $\frac { \sqrt { 3 } } { 2 }$
(3) $\frac { 1 - \sqrt { 3 } } { 2 }$
(4) $\frac { 1 + \sqrt { 3 } } { 2 }$

The value of $\cot \frac { \pi } { 24 }$ is:
(1) $\sqrt { 2 } + \sqrt { 3 } + 2 - \sqrt { 6 }$
(2) $\sqrt { 2 } + \sqrt { 3 } + 2 + \sqrt { 6 }$
(3) $\sqrt { 2 } - \sqrt { 3 } - 2 + \sqrt { 6 }$
(4) $3 \sqrt { 2 } - \sqrt { 3 } - \sqrt { 6 }$

$16 \sin \left( 20 ^ { \circ } \right) \sin \left( 40 ^ { \circ } \right) \sin \left( 80 ^ { \circ } \right)$ is equal to
(1) $\sqrt { 3 }$
(2) $2 \sqrt { 3 }$
(3) 3
(4) $4 \sqrt { 3 }$

The value of $2\sin\frac{\pi}{22}\sin\frac{3\pi}{22}\sin\frac{5\pi}{22}\sin\frac{7\pi}{22}\sin\frac{9\pi}{22}$ is equal to:
(1) $\frac{1}{16}$
(2) $\frac{5}{16}$
(3) $\frac{7}{16}$
(4) $\frac{3}{16}$

The value of $2\sin 12^\circ - \sin 72^\circ$ is
(1) $\frac{\sqrt{5}(1 - \sqrt{3})}{4}$
(2) $\frac{1 - \sqrt{5}}{8}$
(3) $\frac{\sqrt{3}(1 - \sqrt{5})}{2}$
(4) $\frac{\sqrt{3}(1 - \sqrt{5})}{4}$

$\tan \left( 2 \tan ^ { - 1 } \frac { 1 } { 5 } + \sec ^ { - 1 } \frac { \sqrt { 5 } } { 2 } + 2 \tan ^ { - 1 } \frac { 1 } { 8 } \right)$ is equal to:
(1) 1
(2) 2
(3) $\frac { 1 } { 4 }$
(4) $\frac { 5 } { 4 }$

If $0 < x < \frac { 1 } { \sqrt { 2 } }$ and $\frac { \sin ^ { - 1 } x } { \alpha } = \frac { \cos ^ { - 1 } x } { \beta }$, then a value of $\sin \frac { 2 \pi \alpha } { \alpha + \beta }$ is
(1) $4 \sqrt { 1 - x ^ { 2 } } \left( 1 - 2 x ^ { 2 } \right)$
(2) $4 x \sqrt { 1 - x ^ { 2 } } \left( 1 - 2 x ^ { 2 } \right)$
(3) $2 x \sqrt { 1 - x ^ { 2 } } \left( 1 - 4 x ^ { 2 } \right)$
(4) $4 \sqrt { 1 - x ^ { 2 } } \left( 1 - 4 x ^ { 2 } \right)$

The value of $36 \left( 4 \cos ^ { 2 } 9 ^ { \circ } - 1 \right) \left( 4 \cos ^ { 2 } 27 ^ { \circ } - 1 \right) \left( 4 \cos ^ { 2 } 81 ^ { \circ } - 1 \right) \left( 4 \cos ^ { 2 } 243 ^ { \circ } - 1 \right)$ is
(1) 54
(2) 18
(3) 27
(4) 36

The value of $\tan 9^{\circ} - \tan 27^{\circ} - \tan 63^{\circ} + \tan 81^{\circ}$ is $\_\_\_\_$.

For $\alpha, \beta \in \left(0, \frac{\pi}{2}\right)$ let $3\sin(\alpha + \beta) = 2\sin(\alpha - \beta)$ and a real number $k$ be such that $\tan\alpha = k\tan\beta$. Then the value of $k$ is equal to
(1) $-5$
(2) $5$
(3) $\frac{2}{3}$
(4) $-\frac{2}{3}$

If $\sum _ { r = 1 } ^ { 13 } \left\{ \frac { 1 } { \sin \left( \frac { \pi } { 4 } + ( r - 1 ) \frac { \pi } { 6 } \right) \sin \left( \frac { \pi } { 4 } + \frac { r \pi } { 6 } \right) } \right\} = a \sqrt { 3 } + b , a , b \in \mathbf { Z }$, then $a ^ { 2 } + b ^ { 2 }$ is equal to :
(1) 10
(2) 4
(3) 2
(4) 8

$\cos \left( \sin ^ { - 1 } \frac { 3 } { 5 } + \sin ^ { - 1 } \frac { 5 } { 13 } + \sin ^ { - 1 } \frac { 33 } { 65 } \right)$ is equal to:
(1) 1
(2) 0
(3) $\frac { 32 } { 65 }$
(4) $\frac { 33 } { 65 }$

Let the range of the function $f ( x ) = 6 + 16 \cos x \cdot \cos \left( \frac { \pi } { 3 } - x \right) \cdot \cos \left( \frac { \pi } { 3 } + x \right) \cdot \sin 3 x \cdot \cos 6 x , x \in \mathbf { R }$ be $[ \alpha , \beta ]$. Then the distance of the point $( \alpha , \beta )$ from the line $3 x + 4 y + 12 = 0$ is :
(1) 11
(2) 8
(3) 10
(4) 9

Consider vectors $\vec{a}$ and $\vec{b}$ on the coordinate plane satisfying $|\vec{a}| + |\vec{b}| = 9$ and $|\vec{a} - \vec{b}| = 7$. Let $|\vec{a}| = x$, where $1 < x < 8$, and let the angle between $\vec{a}$ and $\vec{b}$ be $\theta$. Using the triangle formed by vectors $\vec{a}$, $\vec{b}$, and $\vec{a} - \vec{b}$, we can express $\cos\theta$ in terms of $x$ as $\frac{c}{9x - x^2} + d$, where $c$ and $d$ are constants with $c > 0$. Let this expression be $f(x)$, with domain $\{x \mid 1 < x < 8\}$. Find $f(x)$ and its derivative. (Non-multiple choice question, 4 points)

$$\frac{\tan 60^{\circ}}{\sin 20^{\circ}} - \frac{1}{\cos 20^{\circ}}$$
Which of the following is this expression equal to?
A) 4
B) 2
C) 1
D) $\frac{\sqrt{3}}{2}$
E) $\frac{1}{2}$

$$\frac{1+\cos 40^{\circ}}{\cos 55^{\circ} \cdot \cos 35^{\circ}}$$
Which of the following is this expression equal to?
A) $\cos 20^{\circ}$
B) $2\cos 20^{\circ}$
C) $4\cos 20^{\circ}$
D) $\cos 40^{\circ}$
E) $2\cos 40^{\circ}$

$\cos \mathrm { x } = \frac { - 4 } { 5 }$ Given this, what is $\cos 2 \mathrm { x }$?
A) $\frac { 3 } { 5 }$
B) $\frac { 5 } { 13 }$
C) $\frac { 12 } { 13 }$
D) $\frac { 24 } { 25 }$
E) $\frac { 7 } { 25 }$

Given that $\alpha , \beta \in \left[ 0 , \frac { \pi } { 2 } \right]$,
$$\sin ( \alpha - \beta ) = \sin \alpha \cdot \cos \beta$$
Which of the following is true?
A) $\alpha = 0$ or $\beta = \frac { \pi } { 2 }$
B) $\alpha = 0$ or $\beta = \frac { \pi } { 4 }$
C) $\alpha = \frac { \pi } { 2 }$ or $\beta = 0$
D) $\alpha = \frac { \pi } { 2 }$ or $\beta = \frac { \pi } { 2 }$
E) $\alpha = \frac { \pi } { 4 }$ or $\beta = 0$

$$\frac { \sin 48 ^ { \circ } } { \sin 16 ^ { \circ } } - \frac { \cos 48 ^ { \circ } } { \cos 16 ^ { \circ } }$$
Which of the following is this expression equal to?
A) $\frac { 3 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 4 } { 3 }$
D) 2
E) 3

For every real number $x$, the number $A$ is defined as $$\sum _ { k = 2 } ^ { 4 } \cos ( 2 k x ) = A$$ Accordingly, $$\sum _ { k = 2 } ^ { 4 } \cos ^ { 2 } ( k x )$$ What is the equivalent of the expression in terms of A?\ A) $A + 2$\ B) $A + 4$\ C) $\frac { \mathrm { A } + 1 } { 2 }$\ D) $\frac { A + 2 } { 2 }$\ E) $\frac { A + 3 } { 2 }$

$$\frac{\cos(2x+y) + \sin(2x-y)}{\cos(2x) + \sin(2x)}$$
Which of the following is the simplified form of this expression?
A) $\cos y - \sin y$ B) $\cos y + \sin y$ C) $\cos x - \sin y$ D) $\sin x - \cos y$ E) $\sin x - \cos x$

For a triangle $ABC$ with side lengths $|BC| = a$ units, $|AC| = b$ units and $|AB| = c$ units,
$$2a^{2} = 2b^{2} + 2c^{2} + 3bc$$
is satisfied. Let $m(\widehat{BAC}) = x$. What is the value of $\tan x$?
A) $-\frac{\sqrt{2}}{3}$ B) $-\frac{\sqrt{3}}{3}$ C) $-\frac{\sqrt{5}}{3}$ D) $-\frac{\sqrt{6}}{3}$ E) $-\frac{\sqrt{7}}{3}$

Let $a$ and $b$ be positive real numbers. In the rectangular coordinate plane, the acute angles that the lines $d_{1}$ and $d_{2}$ shown make with the $x$-axis are $A$ and $B$ respectively, as shown in the figure.
Accordingly, which of the following is the expression for the ratio $\frac{a}{b}$ in terms of $A$ and $B$?
A) $\frac{\tan A}{\tan B}$ B) $\cot A \cdot \cot B$ C) $\cot A - \tan B$ D) $1 + \cot A \cdot \tan B$ E) $1 - \tan A \cdot \cot B$