Exercises: Measuring angles: degrees and radians

Express the following angles given in sexagesimal degrees in radians, using irreducible fractions of $\pi$: $15^\circ$, $135^\circ$, $210^\circ$ and $330^\circ$.

Convert the following angles expressed in radians to sexagesimal degrees: $\frac{\pi}{5}$, $\frac{5\pi}{6}$, $\frac{7\pi}{4}$ and $\frac{11\pi}{3}$.

Reduce the following angles to the first revolution (that is, find their equivalent angle between $0^\circ$ and $360^\circ$) and indicate in which quadrant the terminal side of each lies: $1250^\circ$ and $2460^\circ$.

Find the equivalent positive angle in the first turn (between $0$ and $2\pi$ rad) for the following negative angles: $-\frac{\pi}{4}$ and $-\frac{13\pi}{6}$.

Calculate the central angle subtended by an arc of length $12$ cm in a circle whose radius is $4$ cm. Express the result both in radians and in sexagesimal degrees.

Determine the length of the arc described by the lower end of a $60$ cm long pendulum when it swings through an angle of $\frac{\pi}{8}$ radians.

Calculate the sum of the interior angles of a regular octagon and express the size of each of its interior angles in radians, using fractions of $\pi$.

An angle measures exactly $3$ radians. Without using a calculator, reason whether this angle is greater than, less than or equal to a straight angle ($180^\circ$) and determine which quadrant it is in.

A vinyl record spins on a turntable at a speed of $33$ revolutions per minute (rpm). Calculate its angular velocity in radians per second and determine the angle, in radians, described by a point on its edge in $2.5$ seconds.

On a circular running track, a runner travels along the inside lane, remaining $50$ meters from the center. If they cover a distance of $120$ meters, calculate the angle described relative to the center of the track in radians. Also express this result in degrees, minutes and seconds.

Calculate the smaller angle, expressed in sexagesimal degrees and in radians, formed by the hands of an analogue clock (hour hand and minute hand) at exactly 16:20.

Two cities, A and B, are located on the same terrestrial meridian. The latitude of city A is $41^\circ$ North and the latitude of city B is $28^\circ$ South. Assuming the Earth is a perfect sphere with a mean radius of $6371$ km, calculate the distance along the surface of the Earth separating both cities.

A Ferris wheel has a diameter of $40$ meters and has $24$ cabins evenly spaced along its outer edge. Calculate the distance along the arc of the Ferris wheel separating two consecutive cabins and the angle in radians they form relative to the central axle.

The drive system of an industrial machine consists of two pulleys connected by a taut belt. The driving pulley has a radius of $15$ cm and the driven pulley has a radius of $45$ cm. If the driving pulley turns through an angle of $\frac{7\pi}{2}$ radians, calculate how many radians the driven pulley turns and how many complete revolutions it makes.

An agricultural irrigation sprinkler covers a circular sector. The water jet reaches a maximum distance of $18$ meters and the sprinkler head sweeps through an angle of $140^\circ$. Calculate the total area of land irrigated and the length of the outer arc described by the water.

The three angles of a triangle are in arithmetic progression. Knowing that the largest angle measures $80^\circ$, determine the size of the three angles expressed in radians (using fractions of $\pi$).

A lighthouse projects a beam of light rotating at a constant angular speed of $4$ revolutions per minute. The beam illuminates the surface of the sea up to a maximum distance of $5$ km. Due to the topography of the coast, there is a blind sector of $\frac{2\pi}{5}$ radians inland where the light is not visible. Calculate the total area of the sea illuminated by the lighthouse in each complete turn and the exact time in seconds that the beam of light takes to cross the visible area.

Determine the exact time, between 7:00 and 8:00, at which the hands of an analogue clock (hour hand and minute hand) are perfectly superimposed. Also express the total angle swept through by the hour hand from 7:00 to that precise moment, in radians.

The radius of the wheels of a car is $32$ cm. If the vehicle travels at a constant speed of $90$ km/h, calculate the angular speed of the wheels expressed in radians per second. Afterwards, determine the total angle turned by a wheel over a $15$-minute journey and convert it into the number of complete revolutions.

A gear train consists of three cogwheels A, B and C mounted in series, such that A meshes with B, and B meshes with C. The pitch radii of the wheels are $R_A = 12$ cm, $R_B = 8$ cm and $R_C = 20$ cm. If wheel A rotates through an angle of $150^\circ$ clockwise, determine the angle of rotation of wheel C in radians and specify its direction of rotation.

In Eratosthenes' historic experiment, it was observed that in the city of Syene, the sun's rays struck perpendicular to the ground (angle $0^\circ$) at noon on the summer solstice. At the same time, in Alexandria, located approximately $800$ km north on the same meridian, the rays formed an angle of $\frac{\pi}{25}$ radians with the vertical. Using exclusively these data, deduce the radius of the Earth and the length of its meridian circumference.

A cylindrical roller with a radius of $25$ cm is used to compact the asphalt of a street. When pushed, the roller moves in a straight line, rolling without slipping, and makes exactly $14.5$ complete revolutions. Calculate the horizontal distance covered by the axle of the roller in meters and the total angle turned through by any point on its surface, expressed in sexagesimal degrees.

An initial circular sector has radius $R$ and central angle $\alpha$ (in radians). A new circular sector is designed so that its area is exactly equal to that of the original sector. If, due to design constraints, the radius of the new sector must be triple the original ($3R$), algebraically determine what fraction of the angle $\alpha$ the new sector must have.

A pendulum of length $L = 1$ meter swings describing circular arcs; each swing is the motion from one extreme position to the opposite one. In its first swing, the maximum angle it forms with the vertical is $\frac{\pi}{6}$ radians. Due to friction, in each subsequent swing, the maximum angle reached is reduced by $10\%$ relative to the immediately preceding swing. Setting up the sum of the corresponding infinite geometric series, calculate the total distance traveled by the mass of the pendulum until it stops completely.