Consider the equation

where $0\leq x\leq2\pi$.

A circular sector has perimeter $26\ \text{cm}$ and area $40\ \text{cm}^2$. Its radius is $r\ \text{cm}$ and its central angle is $\theta$ radians.

Consider the equation

where $0\leq x\leq2\pi$.

In triangle $ABC$, side $a=15.0\ \text{m}$, side $b=12.0\ \text{m}$ and angle $A=1.20$ radians.

An alternating voltage is modelled by

where $V$ is measured in volts and $t$ is measured in seconds. Consider $0\leq t\leq0.040$.

In triangle $ABC$, side $a=12.0\ \text{cm}$, side $b=15.0\ \text{cm}$ and angle $A=0.650$ radians.

The temperatures recorded by two sensors during a $12$-hour test are modelled by

and

where temperature is measured in $^\circ\text{C}$ and $0\leq t\leq12$.

A theatre follow-spot illuminates an annular sector of the stage. The inner radius is $4.50\ \text{m}$, the outer radius is $16.0\ \text{m}$ and the angle of the sector is $72^\circ$.

A decorative mosaic has the shape of an annular sector. Its outer radius is $r\ \text{m}$, its inner radius is $(r-2)\ \text{m}$, its central angle is $2.20$ radians and its area is $52.8\ \text{m}^2$.

A curved stained-glass panel is a sector of a circle. Its curved edge has length $9.60\ \text{m}$ and the area of the sector is $28.8\ \text{m}^2$. The two endpoints of the curved edge are joined by a straight support.

The curved edge of a bridge arch is modelled by the minor arc of a circle of radius $30.0\ \text{m}$. The length of this arc is $28.0\ \text{m}$. The region between the arc and its chord is filled with decorative material.

A lampshade panel is cut in the shape of an annular sector. Its inner radius is $18.0\ \text{cm}$, its outer radius is $30.0\ \text{cm}$ and its outer curved edge has length $40.0\ \text{cm}$.

A circular particle detector has outer radius $2.40\ \text{m}$. It contains $150$ identical radial sensors. Each sensor detects a centre-based sector extending from the centre to the detector's outer boundary; its outer arc has length $0.0750\ \text{m}$. Adjacent sensor sectors are separated by equal angular gaps of $0.0100$ radians. There are $150$ such equal gaps, in addition to one remaining angular gap.

The cross-section of water in a horizontal cylindrical tank is modelled by a minor circular segment. The tank has radius $5.50\ \text{m}$ and the curved boundary of the water has length $8.40\ \text{m}$.

The height, $H$, in metres, of a passenger on an observation wheel is modelled by

where $t$ is measured in minutes.

A coastal radar scans a sector of radius $40\ \text{km}$ and central angle $1.10$ radians. After each scan, the sector is rotated anticlockwise through $0.820$ radians. Consecutive scanned sectors therefore overlap.

A tracking marker moves anticlockwise around the unit circle. Initially it is at the point $P$ in the second quadrant, where the horizontal coordinate is $-0.650$. It then moves with constant angular speed $0.720\ \text{rad s}^{-1}$.

During an eight-hour wildlife survey, sound levels at two monitoring stations are modelled by

$S_1(t)=60+8\sin\left(\frac{\pi t}{4}\right)$

and

$S_2(t)=62+6\cos\left(\frac{\pi t}{4}\right)$

where sound level is measured in decibels and $0\leq t\leq8$.

A motorized stage light projects a narrow beam to a wall $25.0\ \text{m}$ from its pivot. Its angular position is modelled by

where $\theta$ is measured in radians and $t$ is measured in seconds.

During a $24$-hour laboratory test, the outputs of two light sensors are modelled by

and

where $0\leq t\leq24$ and output is measured in lux.

Three emergency shelters form triangle $ABC$. The distances opposite angles $A$ and $B$ are $a=27.0\ \text{km}$ and $b=34.0\ \text{km}$ respectively. The measured angle is $A=0.720$ radians.

A point $P$ moves anticlockwise around the unit circle. Its angular position is $\theta$, where $0\leq\theta\leq4\pi$. A sensor is activated whenever $P$ lies on the line

Consider the equation

where $0\leq x\leq4\pi$.

Consider the equation

where $0\leq x\leq2\pi$.

A marker moves anticlockwise around the unit circle with angular position

where $t$ is measured in seconds and $0\leq t\leq8$. Its vertical coordinate is denoted by $y(t)$.

An illuminated display is an annular sector with inner radius $1.80\ \text{m}$ and outer radius $5.20\ \text{m}$. Its boundary has total length $13.6\ \text{m}$. Identical displays are placed around a circular tower. The angular displacement between successive displays is one third of the angle of each display.

Two points $A$ and $B$ lie on a unit circle. The length of the minor chord $AB$ is $1.30$. The angle subtended by this chord at the centre is $\theta$ radians.

Three marine observation stations form triangle $ABC$. The measured values are $a=18.0\ \text{km}$, $b=24.0\ \text{km}$ and $A=0.550$ radians, where side $a$ is opposite angle $A$.

An electrical signal is modelled by

$V(t)=120\sin(40\pi t)+30\cos(40\pi t)$

where $V$ is measured in volts, $t$ is measured in seconds and $0\leq t\leq0.100$.

The vertical coordinates of two markers on a rotating stage are modelled by

$y_A(t)=\sin(0.8t+0.2)$

and

$y_B(t)=\cos(0.8t-0.4)$

where $0\leq t\leq12$. Define $D(t)=y_A(t)-y_B(t)$.

A robotic arm of length $2.40\ \text{m}$ rotates about a fixed pivot. Its angle from the positive horizontal direction is

$\theta(t)=1.10+0.700\sin(0.500t)$

where $0\leq t\leq4\pi$, $t$ is measured in seconds, and the coefficient $0.500$ has units of $\text{s}^{-1}$.

A camera moves anticlockwise on a circular rail of radius $15.0\ \text{m}$. Its angular displacement from its starting point is

$\theta(t)=0.400t+0.250\sin(0.400t)$

where $t$ is measured in seconds and $0\leq t\leq15$.

An acoustic panel has the shape of a circular sector. Its curved edge has fixed length $12.0\ \text{m}$. The radius is $r$ metres and the central angle is $\theta$ radians. Design restrictions require the panel area to be at most $45.0\ \text{m}^2$ and its total perimeter to be at most $26.0\ \text{m}$.

Two ground stations $A$ and $B$ and a drone at $C$ form a triangle. The angle $A$ is $0.900$ radians and the opposite distance is $a=20.0\ \text{km}$. An initial estimate gives $b=18.0\ \text{km}$.

A point rotates around the unit circle with angular position $\theta=0.600t$. A sensor calculates the signal

$S(t)=3\cos^2\theta+2\sin^2\theta$

where $t$ is measured in seconds.

In triangle $ABC$, angle $A=1.05$ radians and side $a=25.0\ \text{m}$. The possible measured value of side $b$ is denoted by $p\ \text{m}$, where $p>0$.

In triangle $ABC$, $A=0.800$ radians, $a=10.0\ \text{cm}$ and the length $b$ may vary.

For a real parameter $k$, consider

$\sin x=k(x-\pi),\qquad 0\leq x\leq2\pi$

The value $x=\pi$ is a solution for every value of $k$.