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.

Lucía has to calculate the area of a landscaped plot shaped like a circular sector, whose radius measures $6\text{ m}$ and whose central angle is $45^\circ$. To solve it, she directly applies the formula $A = \frac{1}{2}r^2\theta$ and writes on her exam:

Upon seeing the result, the teacher tells her to review her calculations, as the complete circle of radius $6\text{ m}$ barely has an area of about $113,1\text{ m}^2$. Identify and explain in detail the conceptual error made by Lucía, justify why the formula used requires the angle to be expressed in radians, and calculate the correct solution both in exact form (in terms of $\pi$) and rounded to two decimal places.

On a circular training track with a radius of $50\text{ m}$, an athlete covers an arc subtending a central angle of $2,4\text{ rad}$.

1. a) Before computing, make an intuitive estimate: knowing that one radian is the central angle whose arc measures exactly the same as the radius, reason without a calculator whether the distance covered by the athlete will exceed $100\text{ m}$ and whether it will be greater or less than $150\text{ m}$.
2. b) Taking into account that $1\text{ rad} \approx 57,3^\circ$ (almost $60^\circ$), estimate mentally the approximate value of the angle in sexagesimal degrees.
3. c) Now calculate with precision the length of the arc covered and the exact value of the angle in sexagesimal degrees (rounded to two decimal places). Compare the results with your previous estimates and assess with reasoning whether your initial approximation was accurate.

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 $50$ metres from the centre. If they cover a distance of $120$ metres, calculate the angle described relative to the centre 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 $40$ metres 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 $18$ metres 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 $\pi$).

To water a grass lawn in a public park, an oscillating sprinkler is used with a range of $8\text{ m}$ that sweeps an angle of $135^\circ$. Two maintenance technicians calculate the watered area following different approaches:

• Method A (Marcos): Sets up a direct proportion based on classical geometry in degrees: $\frac{A}{135^\circ} = \frac{\pi\cdot 8^2}{360^\circ}$.

• Method B (Sara): First converts the angle to radians as an irreducible fraction of $\pi$ and then applies the expression $A = \frac{1}{2}r^2\theta$.

1. a) Carry out both procedures step by step and check whether both technicians obtain the same exact numerical result.
2. b) Analyze the advantages and disadvantages of each method. Which of the two is more direct and coherent if you are subsequently asked to calculate the linear speed of the water at the tip knowing that the sprinkler completes this sweep in $3\text{ s}$? Justify your answer using the relationship $v = \omega\cdot r$.

A children's play area shaped like a circular sector is planned to be built, which will be completely fenced along its perimeter (the two straight edges corresponding to the radii and the curved edge of the arc) using exactly $36\text{ m}$ of metal fencing.

1. a) Express the area $A$ of the circular sector exclusively as a function of the radius $r$, eliminating the angle $\theta$ (in radians) using the condition of the total perimeter of the fence.
2. b) If it is decided to set the radius of the area to $r = 9\text{ m}$, determine the central angle $\theta$ that the sector must have (in radians and in sexagesimal degrees). What exact fraction of the full circle does this area represent?
3. c) Solve the inverse problem: if the central angle is required to be $\theta = 2\text{ rad}$, what must the radius measure to consume exactly the $36\text{ m}$ of fence? Prove that, with this value of the central angle, the length of the curved arc is always equal to the sum of the two straight radii, regardless of the total length of fence available.

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 metres 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 $L = 1$ metre 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 travelled by the mass of the pendulum until it stops completely.

In a wind farm located in the province of Albacete, a three-bladed wind turbine has a rotor whose radius (length of each blade) is $65\text{ m}$. The rotor spins at a constant angular speed of $14\text{ rpm}$ (revolutions per minute).

1. a) Model the angular speed $\omega$ of the rotor expressing it in $\text{rad/s}$ (provide both the exact fraction in multiples of $\pi$ and the approximate decimal value to three significant figures).
2. b) Determine the linear speed $v$ at which the outer tip of each blade moves in $\text{m/s}$ and convert it to $\text{km/h}$. Compare this result with reasoned arguments to the maximum speed allowed on a Spanish motorway ($120\text{ km/h}$) and interpret the mechanical stress experienced by the tips of the blades compared to the areas near the axle.
3. c) A maintenance technician installs a vibration sensor at an intermediate point along a blade. If the sensor records a linear speed of exactly $50\text{ km/h}$, calculate at what distance from the rotation axis it is placed and deduce the total arc length (in kilometres) that this sensor will have traced after an uninterrupted working shift of $8\text{ hours}$.

During a flat stage of the Vuelta a España, a rider rides a racing bicycle whose rear wheel has a total outer radius of $35\text{ cm}$ (including the tyre). The transmission system consists of a front chainring of radius $r_1 = 10,5\text{ cm}$ and a rear sprocket of radius $r_2 = 4,2\text{ cm}$, connected by a taut, inextensible chain that does not slip.

1. a) Algebraically deduce the relationship between the angular speed of the chainring $\omega_{\text{plato}}$ and the angular speed of the rear wheel $\omega_{\text{rueda}}$, relying on the fact that the chain transmits the same linear speed to the edges of both gears ($v = \omega_1 r_1 = \omega_2 r_2$). Reason whether this mechanical gear ratio multiplies the angular speed or the cyclist's force.
2. b) The bicycle rolls on the asphalt at a constant forward linear speed of $42\text{ km/h}$. Using a reverse approach, calculate the angular speed of the rear wheel in $\text{rad/s}$ and the cyclist's pedalling cadence expressed in revolutions per minute (rpm).
3. c) If the cyclist covers a straight stretch of $2,1\text{ km}$, calculate the total angle turned by the rear wheel in radians and the total angle turned by the pedal axle in radians. Verify that the ratio between both angles coincides with the transmission ratio deduced in the first section.