5.1 Angles and Their Measurements

Transcription

1 Graduate T.A. Department of Mathematics Dnamical Sstems and Chaos San Diego State Universit November 8, 2011

2 A ra is the set of points which are part of a line which is finite in one direction, but infinite in the other. A B

3 A ra is the set of points which are part of a line which is finite in one direction, but infinite in the other. A B An angle is defined as the union of two ras with a common endpoint, the verte. C Verte A B

4 An angle can be thought of as being formed b rotating one ra awa from a fied ra. The fied ra is the initial side and the rotated ra is the terminal side. verte terminal side α initial side

5 An angle whose verte is the center of a circle is a central angle, and the arc of the circle through which the terminal side moves is the intercepted arc. Intercepted arc Figure: Central Angle

6 An angle in standard position is located in a rectangular coordinate sstem with the verte at the origin and the initial side on the positive -ais. verte terminal side α initial side

7 The measure of an angle α is denoted b m(α), and indicates the amount of rotation of the terminal side from the initial position. It is found using an circle centered at the verte. verte terminal side α initial side The circle is divided into 360 equal arcs and each arc is one degree (1 o ).

8 The degree measure of an angle is the number of degrees in the intercepted arc of a circle centered at the verte. verte terminal side α initial side

9 HUGE NOTE: The degree measure is positive if the rotation is counterclockwise and negative if the rotation is clockwise

10 Quadrantal angles are angles whose terminal side falls along an ais

11 Coterminal angles are angles in standard position that have the same initial side and same terminal side Definition Angles α and β in standard position are coterminal if and onl if there is an integer k such that m(α) = m(β)+k 360

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13 To define an angle with radian measure 1, consider a circle of an radius, r, centered at the origin of the plane. P r θ r A Definition (Radian Measure) If the length of the intercepted arc between points A and P is equal to the radius of the circle then the angle θ has a measure of one radian.

14 How man radial lengths fit into a half circle? About 3.14 of them. θ r P r A P r α r A m(θ) = 1 radian m(α) = 2 radians r P r β r A P 180 o A m(β) = 3 radians π 3.14 radians

15 A radian is a unitless quantit (0,1) (-1,0) (1,0) (0,-1) Figure: The unit circle is defined b the relation = 1 For radian measure of angles we use a unit circle centered at the origin. Since the radius of the unit circle is the real number 1 without an dimension (such as feet or inches), the length of an intercepted arc is a real number without an dimension.

16 A radian is a unitless quantit (-1,0) (0,1) 57.3 o 1 rad (1,0) (0,-1) Figure: The unit circle is defined b the relation = 1 Hence radian measure of an angle is also a real number without an dimension. One radian (abbreviated 1 rad) is the real number 1. as a result of this construction, converting degrees to radians is a wa to non dimensionalize the units of an angle.

17 Radian Measure (0,1) (0,1) α t (-1,0) (1,0) (-1,0) (1,0) (0,-1) (0,-1) m(α) = π 2 rads m(t) = 3π 4 rads Definition (RADIAN MEASURE) The radian measure of an angle in standard position is simpl the length of the intercepted arc on the unit circle.

18 150, 5π 6 135, 3π 4 120, 2π 3 90, π 2 60, π 3 45, π 4 30, π 6 180, π 360, 2π 210, 7π 6 225, 5π 4 315, 7π 4 330, 11π 6 240, 4π 3 270, 3π 2 300, 5π 3

19 Converting degrees to radians using 180 o = π To change Multipl b Eamples degrees to radians radians to degrees ( ) π π 180 o 150 o = 150 o 180 o = 5π o π ( ) π 225 o = 225 o 180 o = 5π 4 7π 4 = 7π 4 ( ) 180 o = 315 o π π 3 = π ( ) 180 o = 60 o 3 π

20 Common Angles Multiples of are called Eamples (assuming angles are in reduced form) π 2 90 o tpes quadrantal angles 3π 2, 11π 2 7π 2, π π 3 π 4 π 6 60 o tpes 45 o tpes 30 o tpes 2π 3, 5π 3 3π 4, 5π 4 5π 6, 7π 6 4π 3, 217π 3 7π 4, 217π 4 11π 6, 217π 6

21 Theorem (arc length) The length s of an arc intercepted b a central angle of θ radians on a circle of radius r is given b s = r θ θ r s A mnemonic device for remembering s = rθ is SRO (Standing Room Onl) [ s Proof : 2πr = θ ] [ ] [ ] 2πs = 2πrθ s = rθ 2π

22 Eample: A circular arc of length 18 feet subtends a central angle of 75 degrees. Find the radius of the circle in feet. Soln: We are given s = 18 and θ = 75 o. First convert 75 o to radians. θ = 75 o ( ) π 180 o = 5π 12 then use s = rθ and the given values of s and θ to solve for the unknown, r. [ ] [ s = rθ = 18 = 5π ] 12 r Multipling both sides b 12 5π gets the result: ( ) ( )( ) π 18 = 5π 5π 12 r ( ) 12 r = feet 5π

23 Linear and Angular Velocit Suppose that a point is in motion on a circle. (eg. a piece of gum stuck on the tire of a moving car): The distance traveled b the tire in some unit of time is the length of an arc on the circle. The linear velocit of the tire is the rate at which that distance is changing. An angle is formed b the initial and terminal positions of the ra over some unit of time. The angular velocit of the tire is the rate at which that angle is changing. We let the letter v represent linear velocit for a point in circular motion and the Greek letter ω (omega) represent angular velocit.

24 Definition ( Linear Velocit and Angular Velocit) Suppose a point is in constant motion on a circle of radius r (e.g., a piece of gum on a tire moving at 55 mph). Then its linear velocit (constant speed) is v = s t = arclength time where s is the arc length determined b s = rθ (standing room onl). An angle, denoted θ, is formed b the initial and terminal positions of the ra over some unit of time. The angular velocit of the point is ω = θ t = radians time

25 Using the previous definitions v = s t, ω = θ, and the arc length t formula s = rθ (standing room onl) it is eas for one to see that v = s t = rθ ( ) θ t = r = rω t Theorem (Linear Velocit in Terms of Angular Velocit ) If v is the linear velocit of a point on a circle of radius r, and ω (omega) is its angular velocit, then v = rω

26 Eample 11 What are the angular velocit in radians per second and the linear velocit in miles per hour of the tip of a 22-inch lawnmower blade that is rotating at 2500 revolutions per minute?

27 Eample 11 What are the angular velocit in radians per second and the linear velocit in miles per hour of the tip of a 22-inch lawnmower blade that is rotating at 2500 revolutions per minute? Use the fact that 2π radians = 1 revolution and 60 seconds = 1 minute to find the angular velocit, ω. ω = 2500 rev min = 2500 rev min 2π rad 1 rev 1 min rad/sec 60 sec

28 Eample 11 What are the angular velocit in radians per second and the linear velocit in miles per hour of the tip of a 22-inch lawnmower blade that is rotating at 2500 revolutions per minute? Use the fact that 2π radians = 1 revolution and 60 seconds = 1 minute to find the angular velocit, ω. ω = 2500 rev min = 2500 rev min 2π rad 1 rev 1 min rad/sec 60 sec To find linear velocit, use v = rω with r = 11 inches: v = 11 in rad sec in./sec

29 Eample 11 What are the angular velocit in radians per second and the linear velocit in miles per hour of the tip of a 22-inch lawnmower blade that is rotating at 2500 revolutions per minute? Use the fact that 2π radians = 1 revolution and 60 seconds = 1 minute to find the angular velocit, ω. ω = 2500 rev min = 2500 rev min 2π rad 1 rev 1 min rad/sec 60 sec To find linear velocit, use v = rω with r = 11 inches: v = 11 in rad sec Then convert this result to miles per hour: v = in. sec in./sec 1 ft 12 in. 1 mi 3600 sec mi/hr 5280 ft 1 hr