QUESTION 180
The value of $\cos^2 30^\circ + \sin^2 30^\circ$ is
(A) $\frac{1}{2}$
(B) $\frac{\sqrt{3}}{2}$
(C) 1
(D) $\sqrt{3}$
(E) 2

16. Given the function $f ( x ) = ( \sin x + \cos x ) ^ { 2 } + \cos 2 x$
(1) Find the minimum positive period of $f ( x )$;
(2) Find the maximum and minimum values of $f ( x )$ on the interval $\left[ 0 , \frac { \pi } { 2 } \right]$.

17. (This question is worth 12 points) Let the sides opposite to angles $A , B , C$ of $\triangle A B C$ be $a , b , c$ respectively, with $a = b \tan A$. (I) Prove that: $\sin \mathrm { B } = \cos \mathrm { A }$ (II) If $\sin C - \sin A \cos B = \frac { 3 } { 4 }$ and $B$ is an obtuse angle, find $A$, $B$, and $C$.

17. (This question is worth 12 points) Let the sides opposite to angles $A, B, C$ of $\triangle ABC$ be $a, b, c$ respectively. Given $a = b \tan A$ and $B$ is an obtuse angle. (I) Prove: $\mathrm { B } - \mathrm { A } = \frac { \pi } { 2 }$ (II) Find the range of $\sin \mathrm { A } + \sin \mathrm { C }$.

17. (12 points) In $\triangle \mathrm { ABC }$, the sides opposite to angles $\mathrm { A } , \mathrm { B }$, C are $a , b , c$ respectively. The vector $\vec { m } = ( a , \sqrt { 3 } b )$ is parallel to $\vec { n } = ( \cos \mathrm { A } , \sin \mathrm { B } )$. (I) Find A; (II) If $a = \sqrt { 7 } , b = 2$, find the area of $\triangle \mathrm { ABC }$.

17. (12 points) In $\triangle A B C$, the angles $A , B , C$ have opposite sides $a , b , c$ respectively. Given $( \sin B - \sin C ) ^ { 2 } = \sin ^ { 2 } A - \sin B \sin C$.
(1) Find $A$;
(2) If $\sqrt { 2

17. (12 points) In $\triangle A B C$ , let the sides opposite to angles $A , B , C$ be $a , b , c$ respectively. Given $( \sin B - \sin C ) ^ { 2 } = \sin ^ { 2 } A - \sin B \sin C$ .
(1) Find $A$ ;
(2) If $\sqrt { 2 } a + b = 2 c$ , find $\sin C$ .

Given $\sin \theta + \sin \left( \theta + \frac { \pi } { 3 } \right) = 1$, then $\sin \left( \theta + \frac { \pi } { 6 } \right) =$
A. $\frac { 1 } { 2 }$
B. $\frac { \sqrt { 3 } } { 3 }$
C. $\frac { 2 } { 3 }$
D. $\frac { \sqrt { 2 } } { 2 }$

Given $\alpha \in ( 0 , \pi )$ and $3 \cos 2 \alpha - 8 \cos \alpha = 5$, then $\sin \alpha =$
A. $\frac { \sqrt { 5 } } { 3 }$
B. $\frac { 2 } { 3 }$
C. $\frac { 1 } { 3 }$
D. $\frac { \sqrt { 5 } } { 9 }$

18. (12 points) Let the sides opposite to angles $A$, $B$, $C$ of $\triangle A B C$ be $a$, $b$, $c$ respectively. Given that $\frac { \cos A } { 1 + \sin A } = \frac { \sin 2 B } { 1 + \cos 2 B }$.
(1) If $C = \frac { 2 \pi } { 3 }$, find $B$;
(2) Find the minimum value of $\frac { a ^ { 2 } + b ^ { 2 } } { c ^ { 2 } }$ .

``$\sin^{2} \alpha + \sin^{2} \beta = 1$'' is ``$\cos \alpha + \cos \beta = 0$'' a
A. sufficient but not necessary condition
B. necessary but not sufficient condition
C. necessary and sufficient condition
D. neither sufficient nor necessary condition

Let the function $f(x) = 5\cos x - \cos 5x$.
(1) Find the maximum value of $f(x)$ on $\left[0, \frac{\pi}{4}\right]$.
(2) Given $\theta \in (0, \pi)$ and $a$ is a real number, prove that there exists $y \in [a - \theta, a + \theta]$ such that $\cos y \leq \cos \theta$.
(3) If there exists $\varphi$ such that for all $x$, $5\cos x - \cos(5x + \varphi) \leq b$, find the minimum value of $b$.

(17 points)
(1) Find the maximum value of the function $f(x) = 5\cos x - \cos 5x$ on the interval $\left[0, \frac{\pi}{4}\right]$.
(2) Given $\theta \in (0, \pi)$ and $a \in \mathbf{R}$, prove that there exists $y \in [a - \theta, a + \theta]$ such that $\cos y \leq \cos \theta$.
(3) Let $b \in \mathbf{R}$. If there exists $\varphi \in \mathbf{R}$ such that $5\cos x - \cos(5x + \varphi) \leq b$ holds for all $x \in \mathbf{R}$, find the minimum value of $b$.

Let $\theta \in [-\pi, \pi]$.
a. Show that $\cos(\theta) \geq 1 - \frac{\theta^2}{2}$.
b. Show that $\left|\frac{e^{i\theta} - (1-p)}{p}\right| \leq \exp\left(\frac{1-p}{2p^2} \cdot \theta^2\right)$.
Hint. One may calculate $\left|\frac{e^{i\theta} - (1-p)}{p}\right|^2$.

Show the inequality $t\cos(t) \leqslant \sin(t)$, for every $t$ in $[0, \pi/2]$.

Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
Show that for all $x \in \mathbb{R} \backslash \mathbb{Z}$, we have $$f\left(\frac{x}{2}\right) + f\left(\frac{1+x}{2}\right) = 2f(x).$$

Let $f(x) = \pi \operatorname{cotan}(\pi x)$. Show that for all $x \in \mathbb{R} \backslash \mathbb{Z}$, we have $$f\left(\frac{x}{2}\right) + f\left(\frac{1+x}{2}\right) = 2f(x)$$

Let $x \in [ 0,1 [$ and $\theta \in \mathbf { R }$. Using the function $L$, show that
$$\left| \frac { 1 - x } { 1 - x e ^ { i \theta } } \right| \leq \exp ( - ( 1 - \cos \theta ) x )$$
Deduce that for all $x \in [ 0,1 [$ and all real $\theta$,
$$\left| \frac { P \left( x e ^ { i \theta } \right) } { P ( x ) } \right| \leq \exp \left( - \frac { 1 } { 1 - x } + \operatorname { Re } \left( \frac { 1 } { 1 - x e ^ { i \theta } } \right) \right)$$

Let $x \in \left[ \frac { 1 } { 2 } , 1 [ \right.$ and $\theta \in \mathbf { R }$. Show that
$$\frac { 1 } { 1 - x } - \operatorname { Re } \left( \frac { 1 } { 1 - x e ^ { i \theta } } \right) \geq \frac { x ( 1 - \cos \theta ) } { ( 1 - x ) \left( ( 1 - x ) ^ { 2 } + 2 x ( 1 - \cos \theta ) \right) }$$
Deduce that
$$\left| \frac { P \left( x e ^ { i \theta } \right) } { P ( x ) } \right| \leq \exp \left( - \frac { 1 - \cos \theta } { 6 ( 1 - x ) ^ { 3 } } \right) \quad \text { or } \quad \left| \frac { P \left( x e ^ { i \theta } \right) } { P ( x ) } \right| \leq \exp \left( - \frac { 1 } { 3 ( 1 - x ) } \right)$$

Show that $\sin^5 x + \cos^3 x \geq \sin^3 x + \cos^2 x$ implies the expression equals $1$, and find when equality holds.

Let $\alpha$, $\beta$ and $\gamma$ be the angles of an acute angled triangle. Then the quantity $\tan\alpha\tan\beta\tan\gamma$
(a) Can have any real value
(b) Is $\leq 3\sqrt{3}$
(c) Is $\geq 3\sqrt{3}$
(d) None of the above.

Suppose $x, y \in (0, \pi/2)$ and $x \neq y$. Which of the following statement is true?
(A) $2\sin(x + y) < \sin 2x + \sin 2y$ for all $x, y$.
(B) $2\sin(x + y) > \sin 2x + \sin 2y$ for all $x, y$.
(C) There exist $x, y$ such that $2\sin(x + y) = \sin 2x + \sin 2y$.
(D) None of the above.

If the maximum and minimum values of $\sin ^ { 6 } x + \cos ^ { 6 } x$, as $x$ takes all real values, are $a$ and $b$, respectively, then $a - b$ equals
(A) $\frac { 1 } { 2 }$.
(B) $\frac { 2 } { 3 }$.
(C) $\frac { 3 } { 4 }$.
(D) 1 .

If the interior angles of a triangle $ABC$ satisfy the equality, $$\sin ^ { 2 } A + \sin ^ { 2 } B + \sin ^ { 2 } C = 2 \left( \cos ^ { 2 } A + \cos ^ { 2 } B + \cos ^ { 2 } C \right) ,$$ prove that the triangle must have a right angle.

Suppose $x , y \in ( 0 , \pi / 2 )$ and $x \neq y$. Which of the following statements is true?
(a) $2 \sin ( x + y ) < \sin 2 x + \sin 2 y$ for all $x , y$.
(b) $2 \sin ( x + y ) > \sin 2 x + \sin 2 y$ for all $x , y$.
(c) There exist $x , y$ such that $2 \sin ( x + y ) = \sin 2 x + \sin 2 y$.
(d) None of the above.