Compute $$\sin^2 80^\circ -\sin^2 40^\circ +\sqrt{3}\sin 40^\circ \cos 80^\circ $$
Compute $$\sin^2 20^\circ -\sin 5^\circ (\sin 5^\circ +\frac{\sqrt{6}-\sqrt{2}}{2}\cos 20^\circ)$$
Let $\alpha, \beta \in (0, \frac{\pi}{2})$. Show that $\alpha + \beta = \frac{\pi}{2}$ if and only if $$\frac{\sin^4 \alpha}{\cos^2 \beta} + \frac{\cos^4\alpha}{\sin^2\beta} = 1$$
If $\sin\alpha + \sin\beta = \frac{3}{5}$ and $\cos\alpha+\cos\beta=\frac{4}{5}$, compute $\cos(\alpha -\beta)$ and $\sin(\alpha+\beta)$.
Suppose the point $F$ is inside a square $ABCD$ such that $BF=1$, $FA=2$, and $FD=3$, as shown. Find the measurement of $\angle{BFA}$.
An infinite number of equilateral triangles are constructed as shown on the right. Each inner triangle is inscribed in its immediate outsider and is shifted by a constant angle $\beta$.
If the area of the biggest triangle equals to the sum of areas of all the other triangles, find the value of $\beta$ in terms of degrees.
In $\triangle{ABC}$, if $(b+c):(c+a):(a+b)=5:6:7$, compute $\sin{A}:\sin{B}:\sin{C}$.
Let quadrilateral $ABCD$ inscribe in a circle. If $\angle{CAD}=30^\circ$, $\angle{ACB}=45^\circ$, and $CD=2$. Find the length of $AB$.
In $\triangle{ABC}$, if $\sin{A}:\sin{B}:\sin{C}=4:5:7$, compute $\cos{C}$ and $\sin{C}$.
In $\triangle{ABC}$, if $(a+b+c)(a+b-c)=3ab$, compute $\angle{C}$.
Two people located 500 yards apart have spotted a hot air balloon. The angle of elevation from one person to the balloon is $67^\circ$. From the second person to the balloon the angle of elevation is $46^\circ$. How high is the balloon when it is spotted? (You can use a calculator. Please keep the result to the 2 decimal places.)
In $\triangle{ABC}$, if $(a^2 +b^2)\sin(A-B)=(a^2-b^2)\sin(A+B)$, determine the shape of $\triangle{ABC}$.
Let $\triangle{ABC}$ be an acute triangle and $a, b, c$ be the three sides opposite to $\angle{A}, \angle{B}, \angle{C}$ respectively. If vectors $m=(a+c,b)$ and $n=(a-c, b-a)$ satisfy $m\cdot n = 0$,
(1) Compute the measurement of $\angle{C}$.
(2) Find the range of $\sin{A} + \sin{B}$.
In $\triangle{ABC}$, if $a\cos{C} + \frac{c}{2} = b$, (1) compute $\angle{A}$. (2) if $a=1$, find the range of the perimeter o f $\triangle{ABC}$.
Let $BD$ be a median in $\triangle{ABC}$. If $AB=\frac{4\sqrt{6}}{3}$, $\cos{B}=\dfrac{\sqrt{6}}{6}$, and $BD=\sqrt{5}$, find the length of $BC$ and the value of $\sin{A}$.
Let $ABCD$ be inscribed in a circle. If $AB=a, BC=b, CD=c,$ and $DA=d$, show that $$\cos{B} = \frac{a^2 + b^2 -c^2 - d^2}{2(ab+cd)}$$
There are four points on a plane as shown. Points $A$ and $B$ are fixed points satisfying $AB=\sqrt{3}$. Points $P$ and $Q$ can move, as long as $AP=PQ=QB=1$. Let $S$ and $T$ be the area of $\triangle{APB}$ and $\triangle{PQB}$, respectively. Find the maximum value of $S^2+T^2$.
Let $R=\frac{7\sqrt{3}}{3}$ be the circumradius of $\triangle{ABC}$. If $\angle{B} = 60^\circ$ and its area $S_{\triangle{ABC}}=10\sqrt{3}$, find the lengths of $a$, $b$, and $c$.
As shown, prove $$\frac{\sin(\alpha+\beta)}{PC}=\frac{\sin{\alpha}}{PB}+\frac{\sin{\beta}}{PA}$$
In $\triangle{ABC}$, $\angle{BAC} = 40^\circ$ and $\angle{ABC} = 60^\circ$. Points $D$ and $E$ are on sides $AC$ and $AB$, respectively, such that $\angle{DBC}=40^\circ$ and $\angle{ECB}=70^\circ$. Let $F$ be the intersection point of $BD$ and $CE$. Show that $AF\perp BC$.
In $\triangle{ABC}$ show that $$\tan nA + \tan nB + \tan nC = \tan nA \tan nB \tan nC$$ where $n$ is an integer.
In $\triangle{ABC}$, if $A:B:C=4:2:1$, prove $$\frac{1}{a}+\frac{1}{b}=\frac{1}{c}$$
In acute $\triangle{ABC}$, $\angle{ACB}=2\angle{ABC}$. Let $D$ be a point on $BC$ such that $\angle{ABC}=2\angle{BAD}$. Show that $$\frac{1}{BD}=\frac{1}{AB}+\frac{1}{AC}$$
As shown.
Let $O$ be a point inside a convex pentagon, as shown, such that $\angle{1} = \angle{2}, \angle{3} = \angle{4}, \angle{5} = \angle{6},$ and $\angle{7} = \angle{8}$. Show that either $\angle{9} = \angle{10}$ or $\angle{9} + \angle{10} = 180^\circ$ holds.
