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5.1 Radians and Angle Measure

5.1 Radians and Angle Measure

5.1 and Measure

To describe the positions of places on the Earth, cartographers use a grid of that are north-south and east-west. The circles through the poles are called meridians of . The circles parallel to the are called parallels of latitude. The meridians of longitude each have a that equals the radius of the Earth. The radii of the parallels of latitude vary with the latitude.

INVESTIGATE & INQUIRE

1. The angle θ is said to be subtended at the centre of the by the arc AB. A Suppose that the length of the arc AB r θ equals the radius of the circle, r. Estimate r the measure of ∠θ. Justify your estimate. r B

2. In Ontario, the city of London and the town of Moosonee N Moosonee have approximately the same longitude, 81°W. London The diagram shows a circular cross section of the Earth through 81°W and 99°E. Determine the measure of each of W E the following . C a) ∠ECN b) ∠ECW c) reflex ∠ECS 3. What fraction of the of the circle is the length of the arc that subtends each angle in question 2? S 4. If the radius of the circle is r, what is the circumference of the circle in terms of r and π? 5. What is the length, in terms of π and r, of the arc that subtends each angle in question 2? 6. The radius of the Earth is approximately 6400 km. What is the length of the three arcs in terms of π?

328 MHR • Chapter 5 7. Describe a method for finding the length of an arc of a circle using the radius and the fraction of the circumference that equals the . 8. The latitude of London is 43°N, and the latitude of Moosonee is 51.4°N. Determine the distance between London and Moosonee along meridian 81°W, to the nearest kilometre. 9. If the length of an arc on a meridian of longitude is equal to the radius of the Earth, about 6400 km, express 6400 km the measure of the angle subtended at the centre in terms of π. 6400 km 6400 km

10. If the radius of a circle is r and the length of an arc is r, express the measure of the angle subtended at the centre a) in terms of π b) to the nearest tenth of a 11. Compare your answer in question 10b) with your estimate in question 1. Explain any difference between the two values.

Recall that an angle is in standard when its vertex is at the y origin and the initial arm is fixed on the positive x-axis. The terminal terminal arm rotates about the origin. If the direction of rotation is arm counterclockwise, the measure of the angle is positive. 0 x y y

130° 245° 0 x 0 x

When the terminal arm rotates, it may make one or more revolutions. When the terminal arm makes exactly one revolution counterclockwise, the angle has a measure of 360°. y y

480° 0 x 0 x 360°

5.1 Radians and Angle Measure • MHR 329 The degree is a commonly used unit for the measures of angles in trigonometric applications. In the late 1800s, mathematicians and physicists saw the need for another unit, called a , that would simplify some calculations. One radian is the measure of the angle subtended at the centre of a circle by an arc equal in length to r θ the radius of the circle. length of arc = r r

r r r r r θ r θ θ 2 3 1 r

r r r r r θ = θ = θ = 1 1 2 2 rad 3 3 rad The abbreviation rad means radians. The arc lengths in the diagrams are r, 2r, and 3r. Note that the arc of length 3r is not θ = θ = θ = So, 1 1 2 2 3 3 a semicircle. = r = 2r = 3r r r r These relationships lead to the following generalization. θ = a or number of radians = arc length r radius

Since θ = a, then a = rθ, θ > 0 r where a is the arc length, r is the radius, and θ is the measure of the angle, in radians. Note that, when angle measures are written in radians, the unit rad is often omitted. For example, you should assume that an angle measure written as 2 is 2 rad, and that an angle measure written as π is π rad.

330 MHR • Chapter 5 EXAMPLE 1 Using a = rθ Find the indicated quantity in each diagram. a) b) c)

r 10 cm 6 cm 3.2 28 cm 2 θ 45 cm

a SOLUTION a a a) a = rθ b) θ = c) r = r θ a = 6 × 2 45 28 = 12 θ = r = = 10 3.2 So, a 12 cm. = 4.5 = 8.75 So, θ = 4.5 rad. So, r = 8.75 cm.

In order to convert from degree measure to radian measure, the relationship between degrees and radians must be established. In degree measure, one revolution is 360°. In radian measure, one revolution is given by π arc length = 2 r radius r = 2π rad So, the relationship between degrees and radians is 2π rad = 360° π = = 180 ° which simplifies to rad 180° or 1 rad π . Also, since 180° =πrad, π then 1° = rad. 180

5.1 Radians and Angle Measure • MHR 331 EXAMPLE 2 Radian Measure to Degree Measure Change each radian measure to degree measure. Round to the nearest tenth of a degree, if necessary. π 5π a) b) c) 2.2 6 4

SOLUTION To change radian measure to degree measure, multiply the number of 180 ° radians by π .

π π 180 ° 5π 5π 180 ° a) rad = b) rad = 6 6 π 4 4 π = 225° = 30°

= 180 ° Estimate c) 2.2 rad 2.2 π 2 × 180 = 120 ⋅ = 126.1° 3

EXAMPLE 3 Degree Measure to Radian Measure in Exact Form Find the exact radian measure, in terms of π, for each of the following. a) 45° b) 60° c) 210°

SOLUTION To change degree measure to radian measure, multiply the number of π degrees by radians. 180 π π π a) 45° = 45 b) 60° = 60 c) 210° = 210 180 180 180 π π π = 45 = 60 = 210 180 180 180 π π 7π = rad = rad = rad 4 3 6

332 MHR • Chapter 5 EXAMPLE 4 Degree Measure to Radian Measure in Approximate Form Change each degree measure to radian measure, to the nearest hundredth. a) 30° b) 240°

SOLUTION π Estimate a) 30° = 30 × 180 30 3 ==90 0.5 ⋅ = 0.52 rad 180 180

π Estimate b) 240° = 240 180 240 × 3 720 ⋅ == 4 = 4.19 rad 180 180

When an object rotates around a centre point or about an axis, like the θ one shown, its angular , w, is the rate at which the central angle, θ, changes. is usually expressed in radians per .

EXAMPLE 5 Angular Velocity

A small electric motor turns at 2000 r/min. The abbreviation r means Express the angular velocity, in radians per second, revolution. as an exact answer and as an approximate answer, to the nearest hundredth.

SOLUTION To complete one revolution, the motor turns through 2π radians. 2000 r/min = 2000 × 2π rad/min 2000 × 2π = rad/s Estimate 60 200 × 3 2000 × 2π 200π = 200 = 3 60 3 ⋅ = 209.44 π The angular velocity is exactly 200 rad/s, or 209.44 rad/s, to the nearest hundredth. 3 5.1 Radians and Angle Measure • MHR 333 Key Concepts • For an angle θ subtended at the centre of a circle of radius r by an arc of length a, number of radians = arc length , or θ = a. radius r •π = = 180 ° rad 180° or 1 rad π To change radian measure to degree measure, multiply the number of 180 ° radians by π . π • Since 180° =πrad, 1° = rad. 180 To change degree measure to radian measure, multiply the number of π degrees by radians. 180 Communicate Your Understanding

1. Describe how you would find the length of arc a in the diagram. 8 cm

60° a

9 2. Describe how you would change π rad to degree measure. 5 3. Describe how you would change 25° to radian measure.

Practise A 5 1. Find the exact number of degrees in the g) 4π h) π 6 angles with the following radian measures. π 11 π π i) j) π a) b) 18 3 3 4 π 7 c) 2π d) k) π l) 5π 2 6 3 3 e) π f) π 4 2 334 MHR • Chapter 5 2. Find the exact radian measure in terms of 6. Determine the approximate measure of π for each of the following angles. ∠θ, to the nearest hundredth of a radian and a) 40° b) 75° c) 10° d) 120° to the nearest tenth of a degree. e) 225° f) 315° g) 330° h) 240° a) b) i) 540° j) 1080° 3. Find the approximate number of degrees, 8 cm θ 9 m θ to the nearest tenth, in the angles with the following radian measures. 11 cm a) 2.5 b) 1.75 c) 0.35 d) 1.25 5 3 17 17 m e) π f) π g) π h) 0.5 7 11 13 c) 8 mm i) 3.14 j) 1.21

4. Find the approximate number of radians, θ to the nearest hundredth, in the angles with 3.5 mm the following degree measures a) 60° b) 150° c) 80° d) 145° e) 230° f) 325° g) 56.4° h) 128.5° i) 255.4° j) 310.5° 7. Determine the length of each radius, r, to the nearest tenth. 5. Determine the length of each arc, a, to a) b) the nearest tenth. a) b) r

r 5.2 1.4 7 cm 4.4 a 1.7 16 cm 63 cm 6.1 cm a c) c) r

5_ π 2 6 _ π a 42 m 3 12.5 m

5.1 Radians and Angle Measure • MHR 335 Apply, Solve, Communicate

8. Find the indicated quantity in each diagram. Round lengths to the nearest tenth of a and the angle measure to the nearest hundredth of a radian. a) b) 18.2 m

a 10.5 m r 1.6 7_ π 6

c)

8.3 m

θ

30.9 m

9. The end of a 100-cm long pendulum swings through an arc of length 5 cm. Find the radian measure of the angle through which the pendulum swings. Express the radian measure in exact form and in approximate form, to the nearest hundredth. 10. Express the radian measure of each angle in an equilateral triangle in exact form and in approximate form, to the nearest hundredth. 11. Electric motor An electric motor turns at 3000 r/min. Determine the angular velocity in radians per second in exact form and in approximate form, to the nearest hundredth. B 12. Discus throw A discus thrower completes 2.5 r before releasing the discus. Express the of the thrower in radian measure in exact form and in approximate form, to the nearest hundredth.

336 MHR • Chapter 5 π 13. An angle has a radian measure of . Find the exact radian measure of 8 a) its supplement b) its complement 14. Geometry The measures of the acute angles in a right triangle are in the ratio 2:3. Write the radian measures of the acute angles in exact form and in approximate form, to the nearest hundredth. 15. Inquiry/Problem Solving How many times as large, to the nearest tenth, is an angle measuring 2 rad as an angle measuring 2°? 16. Geography a) Through how many radians does the Earth on its axis in a leap year? Express your answer in exact form. b) How many does it take the Earth to rotate through an angle of 150°? c) How many hours does it take the Earth to rotate through an angle of 7π rad? 3 17. Application The revolving restaurant in the CN Tower completes 5 of a revolution every . If Shani and Laszlo ate dinner at the 6 restaurant from 19:15 to 21:35, through what angle did their table rotate during the meal? Express your answer in radian measure in exact form and in approximate form, to the nearest tenth. 18. Propeller The propeller of a small plane has an angular velocity of 1100 r/min. Express the angle through which the propeller turns in 1 s a) in degree measure b) in radian measure in exact form and in approximate form, to the nearest radian. 19. Watch Express the angle that the second hand on a watch turns through in 1s a) in degree measure b) in radian measure in exact form and in approximate form, to the nearest hundredth 20. Clock a) Express the angular velocity of the hour hand on a clock in revolutions per . b) Determine the radian measure of the angle that the hour hand turns through from 15:00 on Friday until 09:00 the following Monday. Express your answer in exact form and in approximate form, to the nearest hundredth. 5.1 Radians and Angle Measure • MHR 337 21. Compact discs A music CD rotates inside an audio player at different rates. The angular velocity is 500 r/min when music is read near the centre, decreasing to 200 r/min when music is read near the outer edge. a) When music is read near the centre, through how many radians does the CD turn each second? Express your answer in exact form and in approximate form, to the nearest hundredth. b) When music is read near the outer edge, through how many degrees does the CD turn each second? c) Computer CD-ROM drives rotate CDs at multiples of the values used in music CD players. A 1X drive uses the same angular as a music player, a 2X drive is twice as fast, a 4X drive is four times as fast, and so on. When data are being read near the centre, through how many degrees does a CD turn each second in a 12X drive? a 40X drive? a 50X drive? d) When data are read near the outer edge, through how many radians does a CD turn each second in a 24X drive? Express your answer in exact form and in approximate form, to the nearest radian. 22. Ferris wheel A Ferris wheel with a radius of 32 m makes two revolutions every . a) Find the exact angular velocity, in radians per second. b) If the ride lasts 3 min, how far does a rider travel, to the nearest metre? 23. Weather satellite A weather satellite completes a circular orbit around the Earth every 3 h. The satellite is 2500 km above the Earth, and the radius of the Earth is approximately 6400 km. How far does the satellite travel in 1 min, to the nearest kilometre? 24. Rolling wheel A wheel turns with an angular velocity of 8 rad/s. a) Express the angular velocity to the nearest tenth of a revolution per minute. b) If the radius of the wheel is 20 cm, how far will it roll in 10 s, to the nearest centimetre? 25. Satellite orbit A satellite with a circular orbit has an angular velocity of 0.002 rad/s. a) How long will it take the satellite to complete one orbit, to the nearest second? b) What is the of the satellite, to the nearest tenth of a kilometre per second, if it is orbiting 800 km above the Earth, and the radius of the Earth is approximately 6400 km?

338 MHR • Chapter 5 26. Navigation satellite The signals from a marine navigation satellite, N, can be received by any ship on arc AB, as shown in the diagram. The signal angle is 15°. What is the length of arc AB, to the nearest 100 km? A 6400 km

C 15° N

B

C 27. Trigonometric ratios State the exact value of each of the following. Explain your reasoning. π π a) sin rad b) cos rad 2 3 5π 2π c) sin d) cos 6 3 28. Tire velocity A car is travelling at 100 km/h. The radius of a tire is 36 cm. a) Find the angular velocity of a tire, to the nearest tenth of a revolution per second. b) Through how many radians will the tire turn in 30 s, to the nearest radian? 29. Communication The equator and the Arctic Circle have different latitudes. In relation to the Earth’s axis of rotation, how do the following compare for a person standing on the equator and a person standing on the Arctic Circle? Explain your reasoning. a) the angular velocity, in radians per hour b) the speed, in kilometres per hour 30. A system of measuring angles uses units called gradians. The circle is divided into 400 equal parts. Each part is called a grad. So one complete revolution measures 400 grad. a) What are the and the cosine of 100 grad? b) Express 100 grad in degree measure and in exact radian measure. c) Express one radian in measure in exact form and in approximate form, to the nearest hundredth.

5.1 Radians and Angle Measure • MHR 339 A CHIEVEMENT Check Knowledge/Understanding Thinking/Inquiry/Problem Solving Communication Application The orbit of the Earth around the sun can be approximately modelled by a circle of radius 150 million kilometres. The Earth completes one revolution every 365 days, and it rotates on its axis once every 24 h. The Earth revolves on its axis in a west to east direction. Assume that the Earth is a sphere with radius 6370 km. a) Find the speed of the Earth, as it orbits the sun, in kilometres per hour. b) Find the angular velocity of the Earth, as it orbits the sun, in radians per hour. c) Find the speed, relative to the Earth’s axis, of a point on the equator, in kilometres per hour. d) Find the speed, relative to the Earth’s axis, of the North Pole. e) Find the speed, relative to the Earth’s axis, of Vancouver, which is 49° north of the equator. f) Find the speed, relative to the Earth’s axis, of the top of the CN tower, which is 533 m high and 44° north of the equator.

CAREER CONNECTION Crafts Thousands of Canadians make their living from crafts, such as pottery, glassblowing, textile printing, and making jewelry. Surveys suggest that about 5% of Canadians spend over seven hours a week each on a craft. In some cases, the craft is a hobby, but many people make or supplement their income by working at a craft.

1. Pottery wheel A pottery wheel of radius 16 cm makes 30 r in 10 s. a) Determine the angular velocity of the wheel, in radians per second. b) In 0.1 s, what distance does a point on the edge of the wheel travel, to the nearest tenth of a centimetre.

2. Use your research skills to investigate the following for a craft that interests you. a) how people learn the craft b) the skills and equipment needed c) how math is used in the craft

340 MHR • Chapter 5

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