5.1 Angles and Their Measure. Objectives

Transcription

1 Objectives 1. Convert between decimal degrees and degrees, minutes, seconds measures of angles. 2. Find the length of an arc of a circle. 3. Convert from degrees to radians and from radians to degrees. 4. Find the area of a sector of a circle. 5. Find the linear speed of an object travelling in circular motion. 29 September Kidoguchi, Kenneth

2 Ray & Vertex A ray ( or half-line) is that portion of a line that starts at a point V on the line and extends indefinitely in one direction. The starting point V of the ray is called its vertex. V Ray Line 29 September Kidoguchi, Kenneth

3 Angle, Initial & Terminal Side, Positive & Negative Angles An angle is formed by two rays that share a common vertex. The angle is formed by rotating from the initial side to the terminal side. If the rotation is counterclockwise, the angle is positive. If the rotation is clockwise, the angle is negative. β γ V α Initial Side V Initial Side V Initial Side Positive Angle, α Negative Angle, β Positive Angle, γ N.B.: α β γ but α, β and γ are said to be co-terminal! 29 September Kidoguchi, Kenneth

4 Angle, Initial & Terminal Side, Positive & Negative Angles Definition from Supplemental Packet: The reference angle for an angle in standard position is the positive acute angle formed by the horizontal axis and the terminal side of the angle. β γ V α Initial Side α V Initial Side V Initial Side α Reference Angle, α Reference Angle, α Reference Angle, α N.B.: α β γ but α, β and γ are said to be co-terminal! 29 September Kidoguchi, Kenneth

5 Standard Position y y Terminal side θ Vertex Initial side x Terminal side Vertex θ Initial side x a) θ in standard position; θ > 0 b) θ in standard position; θ < 0 29 September Kidoguchi, Kenneth

6 Quadrants and Quadrantal Angles 29 September Kidoguchi, Kenneth

7 Angles in Degrees 29 September Kidoguchi, Kenneth

8 Drawing an Angle Draw each angle: a) α = 45º b) β =-90º c) θ = 225º d) φ = 405º 29 September Kidoguchi, Kenneth

9 Drawing an Angle Draw each angle: a) α = 45º b) β =-90º c) θ = 225º d) φ = 405º 29 September Kidoguchi, Kenneth

10 1: Conversions Decimal Degrees and DºM S 1 counterclockwise revolution = 360º 1º = 60 minutes = 60 1 = 60 seconds = 60 Example computation: a) Convert θ = 45º to decimal degrees. Round the answer to three decimal places. 29 September Kidoguchi, Kenneth

11 1: Conversions Decimal Degrees and DºM S 1 counterclockwise revolution = 360º 1º = 60 minutes = 60 1 = 60 seconds = 60 Example computation: b) Convert θ = 21.56º to DºM S. Round the answer to the nearest second. 29 September Kidoguchi, Kenneth

12 1: Conversions Decimal Degrees and DºM S 1 counterclockwise revolution = 360º 1º = 60 minutes = 60 1 = 60 seconds = 60 Example computation: a) Convert θ = 45º to decimal degrees. Round the answer to three decimal places. 1º 1' 1º θ = 45º10'15'' = 45º + 10' + 15'' º 60' 60'' 60' b) Convert θ = 21.56º to DºM S. Round the answer to the nearest second. 60' 60'' θ = 21.56º = 21º º = 21º '=21º 33'+0.6' 1º 1' = 21º 33'36'' 29 September Kidoguchi, Kenneth

13 Angles in Radians A central angle is a positive angle whose vertex is at the centre of a circle of radius r. The rays of a central angle subtend (intersect) an arc on the circle. If the arc length is s, then the angle in radians is: arc length θ = = radius If the arc length is s = r, then θ = 1 radian. s r 29 September Kidoguchi, Kenneth

14 2: The Length of an Arc of a Circle s = rθ 3 3 θ (r, 0) Initial Side θ s = rθ s = rθ 1 1 Angle in radians: arc length s θ = = radius r 2πr 1 revolution = = 2π r N.B.: An angle in radians is length over length, hence a dimensionless quantity. For a circle of radius r, a central angle θ in radians subtends an arc of length s such that: s = r θ 29 September Kidoguchi, Kenneth

15 3: Unit Conversions 1 revolution = 2π radians = 360º π radians = 180º π 1degree = radian radian = degree π Example: Convert each angle in degrees to radians: (a) 80 (b) 140 (c) -30 (d) September Kidoguchi, Kenneth

16 3: Unit Conversions 1 revolution = 2π radians = 360º π radians = 180º π 1degree = radian radian = degree π Example: Convert each angle in degrees to radians: (a) 80 (b) 140 (c) -30 (d) 100 πrad 4 (a) 80º = π rad 180º 9 [ ] πrad 1 (c) 30º = π rad 180º 6 [ ] 29 September Kidoguchi, Kenneth

17 3: Unit Conversions Example: Convert each angle in radians to degrees: (a) 2π/3 (b) 5π/6 (c) 3π/5 (d) 2 29 September Kidoguchi, Kenneth

18 3: Unit Conversions Example: Convert each angle in radians to degrees: (a) 2π/3 (b) 5π/6 (c) 3π/5 (d) 2 2π 180º (a) 120º 3 = πrad 5π 180º (b) 150º 6 = πrad 3π 180º (c) 108º 5 = πrad 180º 360º (d) 2 = πrad π 29 September Kidoguchi, Kenneth

19 3: Unit Conversions Degrees 0º 30º 45º 60º 90º 120º 135º 150º 180º Radians 0 π/6 π/4 π/3 π/2 2π/3 3π/4 5π/6 π Degrees 210º 225º 240º 270º 300º 315º 330º 360º Radians 7π/6 5π/4 4π/3 3π/2 5π/3 7π/4 11π/6 2π 29 September Kidoguchi, Kenneth

20 Example: Finding the Distance Between Two Cities The latitude of a location L is the angle formed by a ray drawn from the centre of Earth to the Equator and a ray drawn from the centre of Earth to L. See Figure. Glasgow, Montana, is due north of Albuquerque, New Mexico. Find the distance between Glasgow (48º9 north latitude) and Albuquerque (35º5 north latitude). See Figure 13(b). Assume that the radius of Earth is 3960 miles. 29 September Kidoguchi, Kenneth

21 Example: Finding the Distance Between Two Cities θ s Given: r = 3960 miles θ G = 48º 9 θ A = 35º 5 s θ A = rθ r θ G Let: θ = θ G θ = 13º4' A πrad 1º πrad = 13º + 4' 180º 60' 180º πrad 1º πrad = 3960[ miles] 13º 4' 180º + 60' 180º miles 29 September Kidoguchi, Kenneth

22 4: Area of a Sector In general: θ = θ A A 1 1 For θ 1 = 2π A 1 = π r 2 A A θ = θ = θ = r θ 1 πr 1 2π 2 The area A of the sector of a circle of radius r formed by a central angle θ radians is: A = r θ 29 September Kidoguchi, Kenneth

23 Example: Area of a Sector Find the area of the sector of a circle of radius 5 feet formed by an angle of 40. Round the answer to two decimal places. 29 September Kidoguchi, Kenneth

24 Example: Area of a Sector Find the area of the sector of a circle of radius 5 feet formed by an angle of 40. Round the answer to two decimal places. Method 1: Recipe A 1 2 = r θ π = ( 5ft ) 40º 2 180º 25π ft 2 = ft 29 September Kidoguchi, Kenneth

25 Method 2: Conceptual 5.1 Angles and Their Measure Example: Area of a Sector Find the area of the sector of a circle of radius 5 feet formed by an angle of 40. Round the answer to two decimal places. A = πr ( 5ft) 25π = ft ft 2 2 = π September Kidoguchi, Kenneth

26 5: Linear & Angular Speed for Circular Motion v = linear speed position at time: t > 0 s = v t = ω = s t angular speed θ position at time: t = 0 = θ t v = ωr 29 September Kidoguchi, Kenneth

27 Example: Finding Linear Speed A child is spinning a rock at the end of a 2-foot rope at the rate of 180 revolutions per minute (rpm). Find the linear speed of the rock when it is released. 29 September Kidoguchi, Kenneth

28 Example: Finding Linear Speed A child is spinning a rock at the end of a 2-foot rope at the rate of 180 revolutions per minute (rpm). Find the linear speed of the rock when it is released. v = r ω rev rad = 2[ft] min π rev ft = 720π min ft min v = ωr r 29 September Kidoguchi, Kenneth

29 Linear Speed and Angular Speed Two insects sit on an old vinyl record. A red insect sits at a point that is 1 inch from the centre of the disk and a green insect sits at a point that is 6 inches from the centre. The record rotates such that it completes 33 1 /3 revolutions per minute. Present the analysis to determine: a) the exact value of the angular speed of each insect in radian/sec, b) the linear speed of each insect (rounded to the nearest inch/sec), and c) the total distance (rounded to the nearest inch) travelled by each insect after 3 minutes. 29 September Kidoguchi, Kenneth

30 Linear Speed and Angular Speed Two insects sit on an old vinyl record. A red insect sits at a point that is 1 inch from the centre of the disk and a green insect sits at a point that is 6 inches from the centre. The record rotates such that it completes 33 1 /3 revolutions per minute. Present the analysis to determine: a) the exact value of the angular speed of each insect in radian/sec, b) the linear speed of each insect (rounded to the nearest inch/sec), and 29 September Kidoguchi, Kenneth

31 Linear Speed and Angular Speed Two insects sit on an old vinyl record. A red insect sits at a point that is 1 inch from the centre of the disk and a green insect sits at a point that is 6 inches from the centre. The record rotates such that it completes 33 1 /3 revolutions per minute. Present the analysis to determine: c) the total distance (rounded to the nearest inch) travelled by each insect after 3 minutes. 29 September Kidoguchi, Kenneth

32 Linear Speed and Angular Speed The minute hand of a clock is 6 centimetres long. You are very bored by the lecture, so you watch it rotate for 35 minutes. Present the analysis to exact values for: a) The angle in degrees spanned by the minute hand after 35 minutes. b) The angle in radians spanned by the minute hand after 35 minutes. c) The distance travelled by the tip of the minute hand after 35 minutes. d) The linear speed of the tip of the minute hand in centimetres per minute. e) The angular speed of the tip of the minute hand in radians per minute 29 September Kidoguchi, Kenneth

33 Linear Speed and Angular Speed a) The angle in degrees spanned by the minute hand after 35 minutes. b) The angle in radians spanned by the minute hand after 35 minutes. c) The distance travelled by the tip of the minute hand after 35 minutes. d) The linear speed of the tip of the minute hand in centimetres per minute. e) The angular speed of the tip of the minute hand in radians per minute 29 September Kidoguchi, Kenneth

34 Example: Finding Linear Speed Earth rotates on an axis through its poles. The distance from the axis to a location on Earth 40 north latitude is about miles. Therefore, a location on Earth at 40 north latitude is spinning on a circle of radius miles. Compute the linear speed on the surface of Earth at 40 north latitude. s v = = t = distance travelled elapsed time 2πr T 2π miles ( ) 24 hour miles/hour 29 September Kidoguchi, Kenneth

35 Angles in Radians Another Example Computation A weather satellite orbits the Earth in a circular orbit 500 miles above the Earth's surface. Assume Earth to be a perfect sphere with radius 3960 miles and the satellite has travelled a distance of 600 miles. Present the analysis to find: a) exact values for the satellite's angular displacement in radians and in degrees of arc and b) approximate values for the satellite's angular displacement rounded to the nearest tenth of a radian and to the nearest tenth of a degree. 29 September Kidoguchi, Kenneth