Practice (117)
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Let integer $n\ge 2$, prove $$\sin{\frac{\pi}{n}}\cdot\sin{\frac{2\pi}{n}}\cdots\sin{\frac{(n-1)\pi}{n}}=\frac{n}{2^{n-1}}$$
Among all pairs of real numbers $(x, y)$ such that $\sin\sin x=\sin\sin y$ with $-10\pi \le x, y \le 10\pi$. Oleg randomly selected a pair $(X, Y)$. Compute the probability that $X = Y$.
Joel selected an acute angle $x$ (strictly between 0 and 90 degrees) and wrote the value of $\sin x$, $\cos x$, and $\tan x$ on three different cards. Then he gave those cards to three students, Malvina, Paulina, and Georgian, one card to each, and asked them to figure out which trigonometric function (sin, cos, tan) produced their cards. Even after sharing the values on their cards with each other, only Malvian was able to surly identify which function produced the value on her card. Compute the sum of all possible values that Joel wrote on Marlvina's card.
If the circle \(x^2 + y^2 = k^2\) covers at least one maximum and one minimal of the curve \(f(x)=\sqrt{3}\sin\frac{\pi x}{k}\), find the range of \(k\).
Let $x, y \in [-\frac{\pi}{4}, \frac{\pi}{4}], a \in \mathbb{Z}^+$, and
$$
\left\{
\begin{array}{rl}
x^3 + \sin x - 2a &= 0 \\
4y^3 +\frac{1}{2}\sin 2y +a &=0
\end{array}
\right.
$$
Compute the value of $\cos(x+2y)$
Prove the following identities
\begin{align}
\sin (3\alpha) &= 4\cdot \sin(60-\alpha)\cdot \sin\alpha\cdot \sin(60+\alpha)\\
\cos (3\alpha) &= 4 \cdot\cos(60-\alpha)\cdot \cos\alpha\cdot \cos(60+\alpha)\\
\tan (3\alpha) &= \tan(60-\alpha) \cdot\tan\alpha \cdot\tan(60+\alpha)
\end{align}
Show that
$$\sin^2\alpha - \sin^2\beta = \sin(\alpha + \beta)\sin(\alpha-\beta)$$
$$\cos^2\alpha - \cos^2\beta = - \sin(\alpha + \beta)\sin(\alpha-\beta)$$
Compute $$\sin^410^{\circ} +\sin^450^{\circ}+\sin^470^\circ$$
Simplify $$\sin^2\alpha + \sin^2\Big(\alpha + \frac{\pi}{3}\Big)+\sin^2\Big(\alpha - \frac{\pi}{3}\Big)$$
Let $A (x_1, y_1)$, $B (x_2, y_2)$, and $C (x_3, y_3)$ be three points on the unit circle, and $$x_1 + x_2 + x_3 = y_1+y_2+y_3=0$$ Prove $$x_1^2 +x_2^2+x_3^2=y_1^2+y_2^2+y_3^2=\frac{3}{2}$$
Compute $(1+\tan 1^\circ)(1+\tan 2^\circ)\cdots(1+\tan 44^\circ)(1+\tan 45^\circ)$
Compute $$\cos\frac{2\pi}{7} \cdot \cos \frac{4\pi}{7}\cdot \cos \frac{8\pi}{7} $$
Compute $$\cos\frac{\pi}{2n+1}\cdot\cos\frac{2\pi}{2n+1}\cdots\cos\frac{n\pi}{2n+1}$$
Compute $$\Big(1+\cos\frac{\pi}{5}\Big)\Big(1+\cos\frac{3\pi}{5}\Big)$$
Compute $$\sin^2 10^\circ + \cos^2 40^\circ + \sin 10^\circ \cos 40^\circ$$
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 $\sin(\alpha+\beta)$.
Compute the value of $\sin{18^\circ}$ using regular geometry.
Let $F$ be a point inside $\triangle{ABC}$ such that $\angle{CAF} = \angle{FAB} = \angle{FBC} = \angle{FCA}$, show that the lengths of three sides form a geometric sequence.
Suppose the point $F$ is inside a square $ABCD$ such that $BF=1$, $FA=2$, and $FD=3$, as shown. Find the measurement of $\angle{BFA}$.
If real numbers $a$ and $b$ satisfy $a^2 + b^2=1$, find the minimal value of $a^4 + ab+b^4$.
Alice and Bob live $10$ miles apart. One day Alice looks due north from her house and sees an airplane. At the same time Bob looks due west from his house and sees the same airplane. The angle of elevation of the airplane is $30^\circ$ from Alice's position and $60^\circ$ from Bob's position. Which of the following is closest to the airplane's altitude, in miles?
Let sequence $\{a_n\}$ satisfy the condition: $a_1=\frac{\pi}{6}$ and $a_{n+1}=\arctan(\sec a_n)$, where $n\in Z^+$. There exists a positive integer $m$ such that $\sin{a_1}\cdot\sin{a_2}\cdots\sin{a_m}=\frac{1}{100}$. Find $m$.