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5.1 Radians and Angle Measure

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5.1 Radians and Angle Measure 5.1 Radians and Angle Measure To describe the positions of places on the Earth, cartographers use a grid of circles that are north-south and east-west. The circles through the poles are called meridians of longitude. The circles parallel to the equator are called parallels of latitude. The meridians of longitude each have a radius 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 circle 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 angles. C a) ∠ECN b) ∠ECW c) reflex ∠ECS 3. What fraction of the circumference 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 arc length. 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 degree 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 position 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 radian, 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 rad 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. = ᎏrᎏ = ᎏ2ᎏr = ᎏ3ᎏr r r r These relationships lead to the following generalization. θ = ᎏaᎏ or number of radians = ᎏᎏarc length r radius Since θ = ᎏaᎏ, 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 = ᎏᎏ c) r = ᎏᎏ r θ a = 6 × 2 45 28 = 12 θ = ᎏᎏ r = ᎏᎏ = 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 ᎏᎏarc length = ᎏ2 r radius r = 2π rad So, the relationship between degrees and radians is 2π rad = 360° π = = ᎏ180ᎏ ° which simplifies to rad 180° or 1 rad ΂ π ΃ . Also, since 180° =πrad, π then 1° = ᎏᎏ 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) ᎏᎏ b) ᎏᎏ c) 2.2 6 4 SOLUTION To change radian measure to degree measure, multiply the number of ᎏ180ᎏ ° radians by ΂ π ΃ . π π 180 ° 5π 5π 180 ° a) ᎏᎏ rad = ᎏᎏ ᎏᎏ b) ᎏᎏ rad = ᎏᎏ ᎏᎏ 6 6΂ π ΃ 4 4 ΂ π ΃ = 225° = 30° = ᎏ180ᎏ ° Estimate c) 2.2 rad 2.2΂ π ΃ ᎏ2ᎏ × 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 ᎏᎏ radians. ΂180΃ π π π a) 45° = 45 ᎏᎏ b) 60° = 60 ᎏᎏ c) 210° = 210 ᎏᎏ ΂180΃ ΂180΃ ΂180΃ π π π = ᎏ45ᎏ = ᎏ60ᎏ = ᎏ210ᎏ 180 180 180 π π 7π = ᎏᎏ rad = ᎏᎏ rad = ᎏᎏ 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΂ᎏᎏ΃ × 180 ᎏ30 3 ==ᎏ90 0.5 ⋅ = 0.52 rad 180 180 π Estimate b) 240° = 240 ᎏᎏ ΂180΃ 240 × 3 720 ⋅ ᎏ ==ᎏ 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 velocity, w, is the rate at which the central angle, θ, changes. Angular velocity is usually expressed in radians per second. 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 = ᎏᎏ rad/s Estimate 60 200 × 3 2000 × 2π 200π ᎏ = 200 ᎏᎏ = ᎏ 3 60 3 ⋅ = 209.44 π The angular velocity is exactly ᎏ200 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 = ᎏᎏarc length , or θ = ᎏaᎏ. radius r •π = = ᎏ180ᎏ ° rad 180° or 1 rad ΂ π ΃ To change radian measure to degree measure, multiply the number of ᎏ180ᎏ ° radians by ΂ π ΃ . π • Since 180° =πrad, 1° = ᎏᎏ rad. 180 To change degree measure to radian measure, multiply the number of π degrees by ᎏᎏ 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 ᎏᎏπ 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) ᎏᎏπ 6 angles with the following radian measures. π 11 π π i) ᎏᎏ j) ᎏᎏπ a) ᎏᎏ b) ᎏᎏ 18 3 3 4 π 7 c) 2π d) ᎏᎏ k) ᎏᎏπ l) 5π 2 6 3 3 e) ᎏᎏπ f) ᎏᎏπ 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) ᎏᎏπ f) ᎏᎏπ g) ᎏᎏπ h) 0.5 7 11 13 c) 8 mm i) 3.14 j) 1.21 4.
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