Trigonometric Functions

Which of these angles is coterminal with $400^{\circ}$?

Convert $7\pi/6$ radians to degrees.

What is the angle in radians made by the hour hand of a clock in $3$ hours?

Express $240^{\circ}$ in radians.

A wheel of a bicycle makes $360$ revolutions in one minute. Through how many radians does it turn in one second? The diagram shows the wheel with a radius $r$ and the angle $\theta$ subtended by an arc length $l$ at the center.

Convert $-3$ radians into degrees (approximately).

Find the length of an arc of a circle of radius $5$ cm subtending a central angle measuring $15^{\circ}$.

A railway train is traveling on a curve of $1500\text{ m}$ radius at the rate of $66\text{ km/hr}$. Through what angle (in degrees) will it turn in $10$ seconds? (Use $\pi = \frac{22}{7}$)

Two gears are in mesh. The smaller gear has a radius of $15\text{ cm}$ and the larger gear has a radius of $45\text{ cm}$. If the smaller gear rotates through an angle of $270^{\circ}$, calculate the angle (in radians) through which the larger gear rotates.

An athlete is running on a circular track of radius $100\text{ m}$. If the athlete covers a distance such that the angle subtended at the center is $72^{\circ}$, and then continues for another $20\pi\text{ m}$ along the track, find the total central angle subtended by the entire path in radians.

The graph of $y = \cos x$ is shown below. In which of the following intervals are both $\sin x$ and $\cos x$ negative?

A point $P$ lies on the terminal side of an angle $\theta$ in the third quadrant as shown in the coordinate system. If $\tan \theta = \frac{4}{3}$, what is the value of $\sin \theta + \cos \theta$?

What is the value of $\cos(-\pi)$?

Evaluate $\sin(765^{\circ})$.

For a given angle $x$, the product $\sin x \cdot \cos x$ is positive. In which quadrants could the terminal side of angle $x$ lie?

Consider the unit circle shown. If the point $M$ has coordinates $(\cos \alpha, \sin \alpha)$ and $\alpha$ is in the fourth quadrant such that $\cos \alpha = 0.6$, what is the value of $\sin \alpha$?

If $\sec x = -\frac{13}{5}$ and $x$ lies in the second quadrant, determine the value of $\frac{1 + \tan x}{1 - \cot x}$.

A laser beam is reflected at an angle $\theta$. If $\tan \theta = \frac{4}{3}$ and $\sin \theta < 0$, find the value of $3 \sin \theta + 4 \cos \theta - 5 \tan \theta$.

Determine the quadrant in which the angle $\theta$ lies if both $\frac{\sin \theta}{\cos \theta} < 0$ and $\sin \theta + \cos \theta < 0$ with $|\sin \theta| > |\cos \theta|$.

A satellite trajectory is modeled by an angle $\theta$ where $\cot \theta = -\frac{5}{12}$ and $\sin \theta > 0$. Calculate the value of $\frac{1 - \cos \theta}{1 + \sin \theta}$.

Which of these functions has a range of $(-\infty, \infty)$?

What is the minimum value of $f(x) = \cos(x) + 2$?

The domain of $f(x) = \sec(x)$ is the same as the domain of which function?

In which quadrants is the range of $\sin(x)$ positive?

The range of $f(x) = \sin(x) + \cos(x)$ is:

What is the range of $f(x) = \frac{\tan(x)}{\sec(x)}$?

Determine the domain of $f(x) = \sqrt{\cos(\sin(x))}$.

An oceanographer uses the function $D(\theta) = \frac{4}{2 + \sin \theta}$ to model depth variation. What is the difference between the maximum and minimum depths predicted by this model?

The displacement of a vibrating string is given by $y = 3\sin(x) \cos(x)$. By rewriting this using a double-angle identity, find the range of the displacement $y$.

In a digital filter design, the gain is given by $H(\omega) = |\sin \omega| + |\cos \omega|$. What is the maximum value that the gain $H(\omega)$ can achieve?

Using the identity for $\cos(A-B)$, find the exact value of $\cos(15^{\circ})$. The figure illustrates the subtraction of $30^{\circ}$ from $45^{\circ}$.

The value of $\sin(75^{\circ})$ can be calculated by expressing $75^{\circ}$ as the sum of $45^{\circ}$ and $30^{\circ}$. Based on the unit circle shown, which of the following is the correct value?

Find $\cos 105^\circ$ using sum and difference formulas.

The value of $\tan(A-B)$ if $\tan A = 2$ and $\tan B = 1$ is:

Find the value of $\cos(x + y)$ given that $\sin x = \frac{3}{5}$, $\cos y = \frac{12}{13}$, where $x$ and $y$ are acute angles. The triangles represent the ratios for $x$ and $y$.

Evaluate the expression $\frac{\tan(160^{\circ}) - \tan(100^{\circ})}{1 + \tan(160^{\circ})\tan(100^{\circ})}$. Use the tangent subtraction identity illustrated by the slopes in the diagram.

Consider the function $f(x) = \sin(x) \cos(\frac{\pi}{4}) + \cos(x) \sin(\frac{\pi}{4})$. What is the period of this function, and what is its value at $x = \frac{\pi}{4}$?

Evaluate the expression $\frac{\cos 11^{\circ} + \sin 11^{\circ}}{\cos 11^{\circ} - \sin 11^{\circ}}$. (Hint: Divide numerator and denominator by $\cos 11^{\circ}$ and use the formula for $\tan(45^{\circ} + A)$).

A light ray enters a medium and its angle of refraction $r$ is related to the angle of incidence $i$ such that $\sin(i-r) = \frac{1}{2}$. If $\cos(i-r) = \frac{\sqrt{3}}{2}$, what is the value of $\tan(i-r)$?

In a navigation system, the bearing of a ship changes from angle $\theta_1$ to $\theta_2$. The relative change is expressed as $\frac{\tan \theta_1 - \tan \theta_2}{1 + \tan \theta_1 \tan \theta_2}$. If $\theta_1 = 70^{\circ}$ and $\theta_2 = 10^{\circ}$, find the value of this expression.

What is the principal solution of $\cos x = -\frac{1}{2}$?

Which of the following represents the general solution for $\tan x = \tan \alpha$?

If $\sin x = -\frac{1}{2}$, what is the principal solution in the interval $[0, 2\pi)$?

Find the general solution for $\tan x = 0$.

Find the general solution of $2\sin^2 x + \sin^2 2x = 2$.

Solve $\sin 2x - \sin 4x + \sin 6x = 0$.

What is the general solution for $\tan x + \cot x = 2$?

Two harmonic oscillators are in phase when $\tan(3x) = \tan(x)$. Determine the general solution for $x$.

An AC circuit's current is given by $I = \sin(x) + \sqrt{3}\cos(x)$. The circuit breaker trips when $I = 2$. Find the general solution for the phase $x$.

A particle moves in a circle. Its angular position $\theta$ satisfies $\sin(2\theta) + \cos(2\theta) = 1$. If $\theta \in [0, 2\pi]$, how many distinct positions can the particle occupy?

In the relation between real numbers and radians, if $x$ is a real number, which identity is always true for any point on the unit circle?

In a unit circle, if the arc length $l$ is $\frac{\pi}{3}$, what is the value of the real number $x$ representing the radian measure?

A real number $x$ is equivalent to an angle of $x$ radians. If $x = \pi$, find the value of $\sin(x)$.

What is the real number value associated with a point on the unit circle that has coordinates $(0, -1)$ after one half-revolution in the clockwise direction from $(0, 1)$?

In the study of trigonometric functions, we often establish a relationship between radian measure and real numbers. Consider a unit circle centered at the origin $O(0,0)$ on a Cartesian plane. Let a line $L$ be tangent to the circle at point $A(1, 0)$. If we treat this tangent line as a real number line with $A$ representing $0$, then every real number $t$ on this line can be wrapped around the circle. If a point $P$ on the circle corresponds to the real number $t = \frac{2\pi}{3}$, find the radian measure of the angle $\angle AOP$.

A particle moves on a unit circle from $(1, 0)$ for a distance of $3\pi/2$ in the counter-clockwise direction and then $\pi$ in the clockwise direction. What is its final real number position on the circle?

For a real number $x$, the value of $\tan(x)$ is undefined when the corresponding point on the unit circle has an $x$-coordinate of:

A rotating beacon in a lighthouse is synchronized with a digital clock. The beacon's angle $\theta$ (in radians) is numerically equal to the real number $t$ representing the time in seconds elapsed since midnight. A sensor at the lighthouse is triggered when $\sin(t) = \cos(t)$. Given that the beacon has completed more than one full revolution but less than one and a half revolutions, what is the exact real number value of $t$ when the sensor triggers?

A particle moves along a unit circle centered at the origin, representing the real number line wrapped around the circle. If the particle starts at the point $(1, 0)$ and moves in a counter-clockwise direction to a point $P$ such that the length of the arc traversed is exactly $5$ units, determine which quadrant the point $P$ lies in and the approximate value of the real number $x$ (in radians) that corresponds to this position.

A circular wire of radius $3$ cm is cut and bent so as to lie along the circumference of a hoop whose radius is $48$ cm. Find the real number representing the angle in degrees which it subtends at the center of the hoop.

Convert $\frac{3\pi}{4}$ radians into degrees.

The measure of a straight angle in radians is:

Find the radian measure of $360^{\circ}$.

Convert $90^{\circ}$ to radians.

In a unit circle, an arc of length $l$ subtends an angle $\theta$ at the center. If the arc length is $\frac{3\pi}{4}$ units, what is the degree measure of the angle $\theta$?

A rotating arm in a piece of machinery sweeps through an angle of $150^\circ$. What is the equivalent measure of this angle in radians?

Find the radius of the circle in which a central angle of $60^{\circ}$ intercepts an arc of length $37.4$ cm (use $\pi = \frac{22}{7}$).

In a precision clock, the minute hand has a length of $15$ cm. Consider the movement of the tip of the minute hand between 9:00 AM and 9:25 AM. Let $L$ be the distance traveled by the tip and $\alpha$ be the angle in radians swept by the hand. If we define a value $X$ such that $X = \frac{180 \times L}{15 \pi}$, find the value of $X$ in degrees, representing the equivalent angular displacement.

A satellite is orbiting Earth in a perfectly circular orbit. Over a specific interval of time, the satellite traverses an arc length of $s = 4400$ km. If the radius of the orbit from the center of the Earth is $R = 7000$ km, calculate the angle $\theta$ subtended by this arc at the center of the Earth in degrees. (Use $\pi \approx \frac{22}{7}$)

If the arcs of the same length in two circles subtend angles of $65^{\circ}$ and $110^{\circ}$ at the center, find the ratio of their radii.