Trigonometry

In the following coordinate system, a point $P$ lies on the terminal side of an angle $\theta$ in standard position. If the terminal side passes through the point $(\sqrt{3}, 1)$, what is the measure of the angle $\theta$ in radians?

A circle has a radius of $12$ cm. What is the length of an arc that subtends an angle of $60^{\circ}$ at the center of the circle?

Express $270^\circ$ in radians.

An angle in a circle of radius $5$ cm is $1$ radian. What is the length of the arc?

Express in degrees the angle which subtends at the centre of a circle of radius $3$ yards an arc of length $1$ foot.

If the angular diameter of the moon be $30'$, how far from the eye must a coin of diameter $2$ cm be kept so as to hide the moon?

The perimeter of a certain sector of a circle is equal to half that of the circle of which it is a part. Find the circular measure of the angle of the sector.

Express the angle $-40^{\circ} 30'$ in radian measure.

A satellite orbits the Earth along the equator at an altitude where the radius of the orbit is 42,000 km. If the satellite travels an arc distance of 11,000 km, find the angle subtended at the center of the Earth in degrees. (Use $\pi = \frac{22}{7}$)

Find the magnitude in radians and degrees of the interior angle of a regular octagon.

Find the value of $\sin(-\frac{\pi}{2})$ from the given graph of the sine function.

The graph shown represents $y = \sin(x)$ shifted horizontally. What is the phase shift?

Which of the following points lies on the graph of $y = \tan(x)$?

What is the maximum value reached by the function $y = 5\sin(x) - 2$?

Looking at the graph of $y = \sin(x)$ and $y = \cos(x)$, at which point in $[0, \pi]$ do they intersect?

If the graph shows $y = A\cos(Bx) + D$, what is the value of $D$?

Determine the phase shift of the function $y = \cos(x + \frac{\pi}{4})$ as shown in the graph.

A coastal engineer is monitoring a tidal wave in a harbor. The height of the water level, $h(t)$ in meters, relative to the mean sea level, is modeled by the function $h(t) = a \sin(b(t - c)) + d$, where $t$ is the time in hours after midnight. A sensor captures a specific portion of the tidal graph as shown below. Based on the observation that the tide reaches a maximum height of $3$ meters at 3:00 AM and a minimum height of $-3$ meters at 9:00 AM, which of the following represents the correct trigonometric function for this tidal cycle?

Determine the sum of all values of $x$ in the interval $[0, 2\pi]$ such that $2 \sin^2(x) - 3 \sin(x) + 1 = 0$.

A ferris wheel with a radius of $10$ m completes one revolution every $40$ seconds. The bottom of the wheel is $2$ m above the ground. If a rider starts at the bottom ($t=0$), which function models their height $h(t)$ after $t$ seconds?

Evaluate $\sin 15^{\circ} \cos 75^{\circ} + \cos 15^{\circ} \sin 75^{\circ}$ using the sum formula.

Simplify $\cos(A+B) \cos(A-B) + \sin(A+B) \sin(A-B)$.

If $A+B = 225^{\circ}$, evaluate $\frac{\tan A + \tan B}{1 - \tan A \tan B}$.

Determine $\cos(n+1)x \cos(n-1)x + \sin(n+1)x \sin(n-1)x$ using the subtraction formula for cosine.

Simplify $\frac{\sin(A-B)}{\cos A \cos B} + \frac{\sin(B-C)}{\cos B \cos C} + \frac{\sin(C-A)}{\cos C \cos A}$.

If $\tan A = k$ and $\tan B = \frac{1-k}{1+k}$, find the value of $A+B$.

In a triangle, if $\sin A \cos B + \cos A \sin B = \frac{\sqrt{3}}{2}$, find the measure of angle $C$.

A surveyor measures the slopes of two hills as $\tan A = \frac{1}{2}$ and $\tan B = \frac{1}{3}$. Calculate the cosine of the angle between the two hills, given by $\cos(A + B)$.

In a geometry problem, it is given that $\tan A = \frac{m}{m+1}$ and $\tan B = \frac{1}{2m+1}$. Find the sum of angles $A + B$.

A mechanical linkage follows the path defined by $\cos(45^{\circ} - A) - \sin(45^{\circ} + A)$. Simplify this expression to find the net displacement of the linkage.

Evaluate $\frac{1 - \cos 60^{\circ}}{\sin 60^{\circ}}$.

The expression $3 \sin A - 4 \sin^3 A$ is equal to which of the following?

If $\cos A = \frac{4}{5}$, find $\sin \frac{A}{2}$ (assume $A$ is acute).

Identify the sub-multiple angle formula for $\sin \theta$.

In the following coordinate plane, the value of the function $g(x) = 2\sin(x)\cos(x)$ is plotted. If $\sin(A) = \frac{3}{5}$ and $A$ is an acute angle, find the exact value of the expression $2\sin(A)\cos(A)$ which corresponds to a point on this curve.

The graph of the function $f(x) = \cos^2(x) - \sin^2(x)$ is shown below. What is the period of this function, and which trigonometric identity is represented by its curve?

Given $\sin A = \frac{4}{5}$, find $\cos 3A$.

Given a sub-multiple angle identity used in lens curvature, if $\tan(\frac{x}{2}) = t$, find the value of $\frac{1 - t^2}{1 + t^2}$ when $x = 60^\circ$.

In a rotating system, the torque varies as $T = \frac{\sin 3\theta}{\sin \theta} - \frac{\cos 3\theta}{\cos \theta}$. Simplify this expression to find the peak torque constant.

A signal processor calculates the value of $\cos 20^\circ \cos 40^\circ \cos 80^\circ$ to determine the attenuation factor. What is this value?

The general solution of $\tan x = \tan \alpha$ is:

What is the general solution for $\sin x = \sin \alpha$?

Identify the general solution for $\cos x = -1$.

Find the general solution of $\tan x = \sqrt{3}$.

Find the general solution of the trigonometric equation $\cos 3x = \cos 2x$. The graph of $y = \cos 3x - \cos 2x$ is shown below to visualize where the function equals zero.

Identify the general solution of $\sqrt{3}\cos x + \sin x = \sqrt{2}$.

Solve $\sec^2 2x = 1 - \tan 2x$ for the general solution.

A solar panel's alignment is optimized when $\sin x + \cos x = 1 + \sin x \cos x$. Solve for the general solution of the alignment angle $x$.

A thermal expansion model involves the equation $4\sin x \sin 2x \sin 4x = \sin 3x$. For the interval $x \in \mathbb{R}$, which represents the general solution set for the roots?

A particle moves in a circular path such that its position satisfies $\sec^2 2x = 1 - \tan 2x$. Determine the general solution for the angular position $x$.