Exercises: Measurement of angles: degrees and radians

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

Convert to degrees the following angles expressed in radians: $\frac{\pi}{5}$, $\frac{5\pi}{6}$, $\frac{7\pi}{4}$ 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 one is located: $1250^\circ$ and $2460^\circ$.

Find the equivalent positive angle in the first revolution (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 $12$ inches in length in a circle whose radius is $4$ inches. Express the result both in radians and in degrees.

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

Calculate the sum of the interior angles of a regular octagon and express the value 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 in which quadrant it is located.

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 moves along the inner lane keeping $50$ feet from the center. If they cover a distance of $120$ feet, calculate the angle they have described with respect to the center of the track in radians. Also express this result in degrees, minutes and seconds.

Calculate the smaller angle, expressed in degrees and in radians, formed by the hands of an analog clock (hour and minute hands) exactly at 4:20 p.m.

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 that the Earth is a perfect sphere with a mean radius of $6371$ miles, calculate the straight-line distance on the Earth's surface that separates both cities.

A fairground Ferris wheel has a diameter of $40$ feet and has $24$ cabins spaced evenly along its outer edge. Calculate the distance along the arc of the Ferris wheel that separates two consecutive cabins and the angle in radians they form with respect to the central axis.

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

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

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

A maritime lighthouse projects a beam of light that rotates at a constant angular velocity of $4$ revolutions per minute. The beam illuminates the surface of the sea up to a maximum distance of $5$ miles. Due to the coastal topography, 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 revolution and the exact time in seconds that the light beam takes to cross the visible zone.

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

The radius of the wheels of a car is $32$ inches. If the vehicle travels at a constant speed of $90$ mph, calculate the angular velocity of the wheels expressed in radians per second. Subsequently, determine the total angle turned by a wheel in a journey of $15$ minutes and convert it to the number of complete revolutions.

A gear train is composed 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 $R_A = 12$ inches, $R_B = 8$ inches and $R_C = 20$ inches. 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 the ground perpendicularly (angle $0^\circ$) at noon on the summer solstice. At the same time, in Alexandria, located approximately $800$ miles north on the same meridian, the rays formed an angle of $\frac{\pi}{25}$ radians with the vertical. Exclusively from this data, deduce the radius of the Earth and the length of its meridian circumference.

A cylindrical roller of radius $25$ inches is used to compact the asphalt of a street. When pushed, the roller advances in a straight line rolling without slipping and makes exactly $14,5$ complete revolutions. Calculate the horizontal distance traveled by the axis of the roller in feet and the total angle turned by any point on its surface, expressed in degrees.

There is an initial circular sector of radius $R$ and central angle $\alpha$ (in radians). A new circular sector is designed with the objective that its area is exactly equal to that of the original sector. If due to design restrictions the radius of the new sector must be three times the original ($3R$), determine algebraically what fraction of the angle $\alpha$ the new sector must have.

A pendulum of length $L = 1$ foot oscillates describing arcs of a circumference; each oscillation is the motion from one extreme position to the opposite one. In its first oscillation, the maximum angle it forms with the vertical is $\frac{\pi}{6}$ radians. Due to friction, in each subsequent oscillation, the maximum angle reached is reduced by $10\%$ with respect to the immediately preceding oscillation. By 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.