O valor de $\sin(30^\circ) + \cos(60^\circ) + \tan(45^\circ)$ é
(A) 1 (B) $\dfrac{3}{2}$ (C) 2 (D) $\dfrac{5}{2}$ (E) 3

QUESTION 153
The value of $\sin 30^\circ + \cos 60^\circ$ is
(A) $\frac{1}{2}$
(B) $\frac{\sqrt{2}}{2}$
(C) 1
(D) $\frac{\sqrt{3}}{2}$
(E) $\sqrt{2}$

QUESTION 166
The value of $\tan 45^\circ + \cos 0^\circ$ is
(A) 1
(B) 2
(C) $\sqrt{2}$
(D) $1 + \sqrt{2}$
(E) $2\sqrt{2}$

Rays of sunlight are hitting the surface of a lake forming an angle $x$ with its surface, as shown in the figure.
Under certain conditions, one can assume that the light intensity of these rays, on the lake surface, is given approximately by $I(x) = K \cdot \sin(x)$, where $k$ is a constant, and assuming that $x$ is between $0^{\circ}$ and $90^{\circ}$.
When $x = 30^{\circ}$, the light intensity is reduced to what percentage of its maximum value?
(A) $33\%$
(B) $50\%$
(C) $57\%$
(D) $70\%$
(E) $86\%$

The value of $\cos(60^\circ) + \sin(30^\circ)$ is:
(A) 0
(B) $\dfrac{1}{2}$
(C) 1
(D) $\dfrac{3}{2}$
(E) 2

Recall that $\sin^{-1}$ is the inverse function of $\sin$, as defined in the standard fashion. (Sometimes $\sin^{-1}$ is called $\arcsin$.) Let $f(x) = \sin^{-1}(\sin(\pi x))$. Write the values of the following. (Some answers may involve the irrational number $\pi$. Write such answers in terms of $\pi$.)
(i) $f(2.7)$
(ii) $f'(2.7)$
(iii) $\int_0^{2.5} f(x)\, dx$
(iv) the smallest positive $x$ at which $f'(x)$ does not exist.

There is a line $l$ passing through the origin O with slope $\tan \theta$. Let $\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }$ be the feet of the perpendiculars from two points $\mathrm { A } ( 0,2 ) , \mathrm { B } ( 2 \sqrt { 3 } , 0 )$ to line $l$, respectively. What is the value of $\theta$ that maximizes the sum of the distances from the origin O to point $\mathrm { A } ^ { \prime }$ and to point $\mathrm { B } ^ { \prime }$, $\overline { \mathrm { OA } ^ { \prime } } + \overline { \mathrm { OB } ^ { \prime } }$? (Given that $0 < \theta < \frac { \pi } { 2 }$.) [3 points]
(1) $\frac { \pi } { 12 }$
(2) $\frac { \pi } { 6 }$
(3) $\frac { \pi } { 4 }$
(4) $\frac { \pi } { 3 }$
(5) $\frac { 5 } { 12 } \pi$

On the coordinate plane, as shown in the figure, for a point P on the circle $x ^ { 2 } + y ^ { 2 } = 1$, let $\theta \left( 0 < \theta < \frac { \pi } { 4 } \right)$ be the angle that the line segment OP makes with the positive direction of the $x$-axis. Let Q be the point where the line passing through P and parallel to the $x$-axis meets the curve $y = e ^ { x } - 1$, and let R be the foot of the perpendicular from Q to the $x$-axis. Let T be the intersection point of the line segment OP and the line segment QR, and let $S ( \theta )$ be the area of triangle ORT. When $\lim _ { \theta \rightarrow + 0 } \frac { S ( \theta ) } { \theta ^ { 3 } } = a$, find the value of $60 a$. [4 points]

For a natural number $n$, let $a _ { n }$ be the $n$-th smallest $x$-coordinate among the intersection points of the line $y = n$ and the graph of the function $y = \tan x$ in the first quadrant.
What is the value of $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { n }$? [4 points]
(1) $\frac { \pi } { 4 }$
(2) $\frac { \pi } { 2 }$
(3) $\frac { 3 } { 4 } \pi$
(4) $\pi$
(5) $\frac { 5 } { 4 } \pi$

For $\theta$ satisfying $\frac { \pi } { 2 } < \theta < \pi$ and $\sin \theta = \frac { \sqrt { 21 } } { 7 }$, what is the value of $\tan \theta$? [2 points]
(1) $- \frac { \sqrt { 3 } } { 2 }$
(2) $- \frac { \sqrt { 3 } } { 4 }$
(3) 0
(4) $\frac { \sqrt { 3 } } { 4 }$
(5) $\frac { \sqrt { 3 } } { 2 }$

What is the maximum value of the function $f ( x ) = 4 \cos x + 3$? [3 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

For a positive number $a$, there is a function $$f ( x ) = \tan \frac { \pi x } { a }$$ defined on the set $\left\{ x \left\lvert \, - \frac { a } { 2 } < x \leq a \right. , x \neq \frac { a } { 2 } \right\}$. As shown in the figure, there is a line passing through three points $\mathrm { O } , \mathrm { A } , \mathrm { B }$ on the graph of $y = f ( x )$. Let $\mathrm { C }$ be the point other than $\mathrm { A }$ where the line parallel to the $x$-axis passing through point $\mathrm { A }$ meets the graph of $y = f ( x )$. When triangle $\mathrm { ABC }$ is equilateral, what is the area of triangle $\mathrm { ABC }$? (Here, $\mathrm { O }$ is the origin.) [4 points]
(1) $\frac { 3 \sqrt { 3 } } { 2 }$
(2) $\frac { 17 \sqrt { 3 } } { 12 }$
(3) $\frac { 4 \sqrt { 3 } } { 3 }$
(4) $\frac { 5 \sqrt { 3 } } { 4 }$
(5) $\frac { 7 \sqrt { 3 } } { 6 }$

The function $$f ( x ) = a - \sqrt { 3 } \tan 2 x$$ has a maximum value of 7 and a minimum value of 3 on the closed interval $\left[ - \frac { \pi } { 6 } , b \right]$. What is the value of $a \times b$? (Here, $a$ and $b$ are constants.) [4 points]
(1) $\frac { \pi } { 2 }$
(2) $\frac { 5 \pi } { 12 }$
(3) $\frac { \pi } { 3 }$
(4) $\frac { \pi } { 4 }$
(5) $\frac { \pi } { 6 }$

For $\theta$ satisfying $\frac{3}{2}\pi < \theta < 2\pi$ and $\sin(-\theta) = \frac{1}{3}$, find the value of $\tan\theta$. [3 points]
(1) $-\frac{\sqrt{2}}{2}$
(2) $-\frac{\sqrt{2}}{4}$
(3) $-\frac{1}{4}$
(4) $\frac{1}{4}$
(5) $\frac{\sqrt{2}}{4}$

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]

The function $f(x) = a\cos bx + 3$ is defined on the closed interval $[0, 2\pi]$ and has a maximum value of 13 at $x = \frac{\pi}{3}$. For the ordered pair $(a, b)$ of two natural numbers $a$ and $b$ satisfying this condition, what is the minimum value of $a + b$? [4 points]
(1) 12
(2) 14
(3) 16
(4) 18
(5) 20

Calculate the exact value of:
$\cos \left( \frac { \pi } { 11 } \right)$

3. Execute the program flowchart shown in the figure, the output value of $S$ is
A. $- \frac { \sqrt { 3 } } { 2 }$ B. $\frac { \sqrt { 3 } } { 2 }$ C. $- \frac { 1 } { 2 }$ D. $\frac { 1 } { 2 }$ [Figure]

4. Among the following functions, the one with minimum positive period $\pi$ and whose graph is symmetric about the origin is
A. $\mathrm { y } = \cos \left( 2 x + \frac { \pi } { 2 } \right)$
B. $y = \sin \left( 2 x + \frac { \pi } { 2 } \right)$
C. $y = \sin 2 x + \cos 2 x$
D. $y = \sin x + \cos x$

5. Among the following functions, the odd function with minimum positive period $\pi$ is
(A) $y = \cos \left( 2 x + \frac { \pi } { 2 } \right)$
(B) $y = \sin \left( x 2 + \frac { \pi } { 3 } \right)$
(C) $y = \sin z + \cos$ $y = \sin x +$

11. The function $f ( x ) = \sin ^ { 2 } x + \sin x \cos x + 1$ has minimum positive period $\_\_\_\_$ and minimum value $\_\_\_\_$.

11. The minimum positive period of the function $f ( x ) = \sin ^ { 2 } x + \sin x \cos x + 1$ is $\_\_\_\_$ , and the decreasing interval is $\_\_\_\_$ .

15. Given $w > 0$, the two closest intersection points of the graphs of $y = 2 \sin w x$ and $y = 2 \cos w x$ have a distance of $2 \sqrt { 3 }$. Then $w =$ $\_\_\_\_$.
III. Solution Questions: This section has 6 questions, for a total of 75 points. Solutions should include written explanations, proofs, or calculation steps.

Given the function $f(x) = \sin^2 x - \sin^2\left(x - \frac{\pi}{6}\right)$, $x \in \mathbb{R}$.
(I) Find the minimum positive period of $f(x)$;
(II) Find the maximum and minimum values of $f(x)$ on the interval $\left[-\frac{\pi}{3}, \frac{\pi}{4}\right]$.

18. A student uses the ``five-point method'' to draw the graph of the function $f ( \mathrm { x } ) = \mathrm { A } \sin ( \omega \mathrm { x } + \varphi ) \left( \omega > 0 , \varphi < \frac { \pi } { 2 } \right)$ during a certain period, creates a table and fills in partial data as follows:

$\omega \mathrm { x } + \varphi$	0	$\frac { \pi } { 2 }$	$\pi$	$\frac { 3 \pi } { 2 }$	$2 \pi$
x		$\frac { \pi } { 3 }$		$\frac { 5 \pi } { 6 }$
$\mathrm {~A} \sin ( \omega \mathrm { x } + \varphi )$	0	5		- 5	0

(I) Please complete the above data, fill in the corresponding positions on the answer sheet, and directly write the analytical expression of the function $f ( \mathrm { x } )$; (II) Shift all points on the graph of $y = f ( \mathrm { x } )$ to the left by $\frac { \pi } { 6 }$ units to obtain the graph of $y = g ( \mathrm { x } )$. Find the center of symmetry of the graph of $y = g ( \mathrm { x } )$ that is closest to the origin O.