Algebraic Trigonometry: all problems

How many solutions does the equation $\sin\left(\frac\pi2\cos x\right) = \cos\left(\frac\pi2\sin x\right)$ have in the closed interval $[0, \pi]$?

Suppose that the roots of the polynomial $P(x) = x^3 + ax^2 + bx + c$ are $\cos\frac{2\pi}{7}$, $\cos\frac{4\pi}{7}$, and $\cos\frac{6\pi}{7}$, where angles are in radians. What is $abc$?

The three side lengths of a triangle are in arithmetic progression with shortest side of length 6. One interior angle measures $120^\circ$. What is the area of the triangle?

Cyrus the frog sits on a flat surface. He jumps, landing 2 feet away. He then chooses a direction at random and again jumps 2 feet. What is the probability that after the second jump Cyrus lands within 1 foot of his starting position?

Usain is walking for exercise by zigzagging across a 100-meter by 30-meter rectangular field, beginning at point $A$ and ending on the segment $\overline{BC}$. He wants to increase the distance walked by zigzagging as shown in the figure below ($APQRS$). What angle $\theta = \angle PAB = \angle QPC = \angle RQB = \cdots$ will produce a length that is 120 meters? (This figure is not drawn to scale. Do not assume that the zigzag path has exactly four segments as shown; there could be more or fewer.)

How many values of $\theta$ in the interval $0 < \theta \le 2\pi$ satisfy

How many real numbers satisfy the equation $\sin(20\pi x) = \log_{20}(x)$?

Let $ABC$ be a triangle inscribed in circle $\omega$. Let the tangents to $\omega$ at $B$ and $C$ intersect at point $P$, and let $\overline{AP}$ intersect $\omega$ at $D$. Find $AD$, if $AB = 5$, $BC = 9$, and $AC = 10$. (Write $AD = \frac mn$ in lowest terms and enter $m + n$.)

The functions $\sin(x)$ and $\cos(x)$ are periodic with least period $2\pi$. What is the least period of the function $\cos(\sin(x))$?

The first two terms of a sequence are $a_1 = 1$ and $a_2 = \frac{1}{\sqrt3}$. For $n \ge 1$,

What is $|a_{2009}|$?

An object moves 8 cm in a straight line from $A$ to $B$, turns at an angle $\alpha$, measured in radians and chosen at random from the interval $(0, \pi)$, and moves 5 cm in a straight line to $C$. What is the probability that $AC < 7$?

The number of $x$-intercepts on the graph of $y = \sin(1/x)$ in the interval $(0.0001, 0.001)$ is closest to

If $\sum_{n=0}^{\infty}\cos^{2n}\theta = 5$, what is the value of $\cos2\theta$?

How many solutions does the equation $\tan(2x) = \cos\left(\frac x2\right)$ have on the interval $[0, 2\pi]$?

Which of the following describes the largest subset of values of $y$ within the closed interval $[0, \pi]$ for which

for every $x$ between $0$ and $\pi$, inclusive?

Given that $A_k = \frac{k(k-1)}{2}\cos\frac{k(k-1)\pi}{2}$, find $|A_{19} + A_{20} + \cdots + A_{98}|$.

Given that $(1 + \sin t)(1 + \cos t) = \frac54$ and

where $k$, $m$, and $n$ are positive integers with $m$ and $n$ relatively prime, find $k + m + n$.

Suppose $a$ is a real number such that the equation

has more than one solution in the interval $(0, \pi)$. The set of all such $a$ can be written in the form $(p, q) \cup (q, r)$, where $p$, $q$, and $r$ are real numbers with $p < q < r$. What is $p + q + r$?