An can be formed by rotating one ray away from a fixed ray indicated by an arrow. The fixed. ray is the and the rotated ray is the.

Transcription

1 Date: 1/29/19 61 Section: Objective: angle angles t their measures An can be formed by rotating one ray away from a fixed ray indicated by an arrow The fixed initial side terminal side ray is the and the rotated ray is the An angle whose vertex is the center of a circle is a central, and the arc of the circle through angle intercepted are standard which the terminal side moves is the An angle in axis is located in a rectangular coordinate system with the vertex at the origin and the initial side on the positive xaxis o_ theta P y beta gamma Degree Measure of Angles The, measure m α, of an angle α is the amount of rotation from the initial side to the terminal side, and is found by using any circle centered at the vertex An angle that forms a complete circle arc is 360 The measure of an degree is the number of degrees in the intercepted arc of a circle centered at the vertex counterclockwise rotation angle, clockwise rotation angle negative positive angle standard An angle in is said to lie in the quadrant where its terminal side lies position

2 Acute angle An angle with a measure between 0 and 90 Obtuse angle An angle with a measure between 90 and 180 Straight angle An angle with a measure of exactly 180 Right angle An angle with a measure of exactly 90 Quadrantal angle An angle in standard position whose terminal side is on an axis Minutes and Seconds Historically, angles were measured by using the degreesminutesseconds (DMS) format, but with calculators it is convenient to have some fractional parts of degrees written in decimal form Each degree is divided into 60 equal parts called minutes (nʹ), and each minute is divided into 60 equal parts called seconds (nʺ) I degree 60 minutes 1 minute seconds 1 degree seconds 60 ' Convert each angle to decimal degrees When necessary, round to four decimal places a) 18 24ʹ b) 10 15ʹ is c) 27 10ʹ20ʺ C 1840 fio 24 ' I s ' I as w 20 ", ji66 ' 3%70056 Convert each angle to degreeminutesseconds format Round to the nearest whole number of seconds a) 566 " b) 174 c) 2834 ok " :* ' 340 6, " I origin Unit circle: A circle of radius that is centered at the 4 ' 6,724 What is the circumference of the unit circle? Ltr 2ND 21T What is the arc length intercepted by a 180 angle (1/2 of the circle)? too What is the arc length intercepted by a 120 angle (1/3 of the circle)? What is the arc length intercepted by a 30 angle? An angle of 225? An angle of 210? r I 5%25 21T (E) 2% T (f) it 32 's 3177 If 2* EkH

3 ' You have just calculated the radian measure of each of these angles The radian measure how times radius around many goes circle part Or of circumference Directed length means that the radian measure is if the rotation of the terminal side is counterclockwise and if the negative clockwise rotation is positive One radian: The angle that intercepts an arc with length equal to the radius of a circle (On the unit circle, one radian is the angle that intercepts an arc of length one) trad 5730 Since the radius of the unit circle is the real number 1, without any dimension (ft, meters, etc), the arc length, and hence the radian measure of an angle, is also a real number without any dimension Thus, one radian (1 rad) is the real number 1 Converting Between Radians and Degrees: Since there are 2π radians in a circle (the circumference of the unit circle is 2π) and 360 in a circle, 2 π radians, 360 or π radians 1800 I Degrees Radians: multiply by Too 1800 Radians Degrees: multiply by 1801T Convert the degree measures to radians: Convert the radian measures to degrees: a) 210 b) 272 a) 5 π it b) 167 radians to too 3 7?n as6 Draw each angle in standard position and determine the quadrant in which each angle lies 5 a) π 4 b) 10 π 3 c) 138 2T Qlll 628 d) / TT

4 Arc Length Often, we want to find the arc length (s) on a circle of radius r, intercepted by an angle α We can do this by determining what fraction of the circle we are looking at, then multiplying by the circumference of the circle Degrees: 2trGFoo) Radians: or *** 2mrµ rt angle must be! in radians 1 A central angle of π 2 intercepts an arc on the surface of the earth that runs from the equator to the North Pole Using 3950 miles as the radius of the earth, find the length of the intercepted arc to the nearest mile )( x620s 2 A wagon wheel has a diameter of 28 inches and an angle of 30 between the spokes What is the length of the arc s (to the nearest hundredth of an inch) between two adjacent spokes? 30, Area of a Sector of a Circle 2iTll4K3oooToJx733in@rwaGE Finding the area (A) of a sector of a circle of radius r with central angle α, is similar to finding the arc length: Determine what fraction of the circle the sector makes up, then multiply by the area of the circle Degrees: TREE) Radians: or r *** angle must be! in 1 Which is bigger: a slice of pizza from a 10ʺ diameter pizza cut into 6 slices, or a slice from a 12ʺ diameter pizza cut into 8 slices? :is radians in " " in '3i a@ 2 A centerpivot irrigation system is used to water a circular field with radius 200 feet In three hours the system waters a sector with a central angle of π 8 What area (in square feet) is watered in that time? w/ # tl2oo5fedx78s4ft@40000a8enteriz

5 360 The terminal side of an angle may be rotated in either a positive or negative direction to get to its final position It can also be rotated for more than one revolution in either direction Two angles with the same terminal side are called coterminal angles start stop at same place but direction Coterminal Angles & or rotation is different To find coterminal angles in degrees, I 360 Find two positive angles and two negative angles that are coterminal with each angle: a) 23 b) Determine whether the angles in each pair are coterminal: a) 128 and 592 b) 8 and yes co terminal not cuter mind Example: Draw each angle in standard position, then name the quadrant in which the terminal side lies a) 255 b) 650 c) QN I, Coterminal Angles: I 21T To find coterminal angles in radians, denominator **** Make sure to find a common Find two positive angles and two negative angles that are coterminal with the given angle π a) b) 6 π 4 c) 7 π 3 d) T # YI 94T 21T 2r l 7 9 IIF 1 4 t

6 3rad Velocity: speed rate of distance overtime Angular Velocity: rate central angle rotates one circle If a point is in motion on a circle through an angle of α radians in time t, then its angular velocity ω is given by α ω t etc) Angular velocity is usually expressed as radians per unit of time (radians/hr, radians/min, radians/sec, Convert 650 rpm (revolutions per minute) to radians per minute (Use the fact that 1 revolution 2 π radians ) 650 rev 40841rad at rad Teu min min min Convert the angular velocity of 1600 rad/hr to rad/sec 1600rad hr n L rai see 44rad See A 24inch lawnmower blade rotates at a rate of 2000 rpm What is the angular velocity in radians per second of a point on the tip of the blade? 2000rad 2pradit#ad2094radmin6osecrev60secseT4reV2Trad8 Find the angular velocity in radians per second for a particle that is moving in a circular path at 4 revolutions per second on a circle of radius 9 ft " see rev rad 251rad Sec Sec

7 rate are length goes around a circle Linear Velocity: If a point is in motion on a circle of radius r through an angle of α radians in time t, then its linear velocity v is s given by v, where s is the arc length determined by s αr t A propeller with a radius of 16 meters is rotating at 1500 revolutions per minute What is the linear velocity in meters per minute for a point on the tip of the propeller? 1500 rev mi 2 # rad m Teu 0 nd min Find the linear velocity in meters per second for a particle that is moving in a circular path at 7 revolutions per second on a circle of radius 15 meters 7 i '2 : What is the linear velocity in miles per hour of the tip of a 20inch lawnmower blade that is rotating at 3000 rpms? tread iii min in hr Linear Velocity in Terms of Angular Velocity: If v is the linear velocity of a point on a circle of radius r, and ω is its angular velocity, then v rω Example: Any point on the surface of the earth (except at the poles) makes one revolution (2π radians) about the axis of the earth in 24 hours So the angular velocity of a point on the earth is 2π 24 or π 12 radians per hour The linear velocity of a point on the surface of the earth depends on its distance from the axis of the earth What is the linear velocity in miles per hour of a point on the equator? (Use 3950 miles as the radius of the earth) Sec 6smm rev 60mi mi hr