§5.1 Angles and Their Measure Objectives 1. Convert ...spot.pcc.edu/~kkidoguc/m112/m112_c05.1.pdf · Find the length of an arc of a circle. 3. ... band gare said to be co-terminal!

§5.1 Angles and Their Measure

Objectives

7 April 2018 1 Kidoguchi, Kenneth

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.


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.

LineRay

V


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.

V Initial Side

Positive

Angle, a

a

V Initial Side V Initial Side

b

Negative

Angle, b

g

Positive

Angle, g

N.B.: a ≠ b ≠ g but a, b and g are said to be co-terminal!


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.

V Initial Side

Reference

Angle, a

a

V Initial Side V Initial Side

b

Reference

Angle, a

g

Reference

Angle, a

a a

N.B.: a ≠ b ≠ g but a, b and g are said to be co-terminal!


Standard Position


a) q in standard position;

q > 0

b) q in standard position;

q < 0

q

Initial side

Terminal side

Vertex x

y

q

Initial side

Terminal side

Vertexx

y


Quadrants and Quadrantal Angles



Angles in Degrees



Drawing an Angle


Draw each angle:

a) a = 45º b) b =-90º c) q = 225º d) f = 405º

m112_c05.1ex1.mw


Drawing an Angle


Draw each angle:

a) a = 45º b) b =-90º c) q = 225º d) f = 405º

m112_c05.1ex1.mw


1: Conversions Decimal Degrees and DºM’S”


1 counterclockwise revolution = 360º

1º = 60 minutes = 60’

1’ = 60 seconds = 60”

Example computation:

a) Convert q = 45º 10’15” to decimal degrees. Round the answer to

three decimal places.





1º = 60 minutes = 60’

1’ = 60 seconds = 60”


b) Convert q = 21.56º to DºM’S”. Round the answer to the nearest

second.





a) Convert q = 45º 10’15” to decimal degrees. Round the answer to

three decimal places.

b) Convert q = 21.56º to DºM’S”. Round the answer to the nearest

second.

1º 1' 1º45º10 '15'' 45º 10 ' 15'' 45.171º

60 ' 60 '' 60 '

q

60 ' 60 ''21.56º 21º 0.56º 21º 33.6'=21º33'+0.6'

1º 1'

21º33'36 ''

q


1º = 60 minutes = 60’

1’ = 60 seconds = 60”


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:

If the arc length is s = r, then q = 1 radian.

arc length

radius

s

rq



2: The Length of an Arc of a Circle

Angle in radians:

arc length

radius

s

rq

(r, 0)

Initial Side

1 1s r q

1q

2 2s r q

2q 3q

3 3s r q

21 revolution 2

r

r

N.B.:

• An angle in radians is

length over length,

hence a dimensionless

quantity.

• For a circle of radius r, a

central angle q in

radians subtends an arc

of length s such that:

s = r q


3: Unit Conversions


1 revolution = 2 radians = 360º ⇒ radians = 180º

1degree radian180

1801radian degree

Example: Convert each angle in degrees to radians:

(a) 80° (b) 140° (c) -30° (d) 100°


3: Unit Conversions


1 revolution = 2 radians = 360º ⇒ radians = 180º

1degree radian180

1801radian degree

Example: Convert each angle in degrees to radians:

(a) 80° (b) 140° (c) -30° (d) 100°

rad 4

(a) 80º rad180º 9

rad 1

(c) 30º rad180º 6


3: Unit Conversions


Example: Convert each angle in radians to degrees:

(a) 2/3 (b) 5/6 (c) 3/5 (d) 2


3: Unit Conversions


Example: Convert each angle in radians to degrees:

(a) 2/3 (b) 5/6 (c) 3/5 (d) 2

2 180º(a) 120º

3 rad

5 180º(b) 150º

6 rad

3 180º(c) 108º

5 rad

180º 360º(d) 2

rad

Degrees 0º 30º 45º 60º 90º 120º 135º 150º 180º

Radians 0 /6 /4 /3 /2 2/3 3/4 5/6

Degrees 210º 225º 240º 270º 300º 315º 330º 360º

Radians 7/6 5/4 4/3 3/2 5/3 7/4 11/6 2


3: Unit Conversions



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.


Example: Finding the Distance Between Two Cities


Given:

r = 3960 miles

qA = 35º 5’

qA

qG

q

s

r

Let:

13º4 '

rad 1º rad13º 4 '

180º 60 ' 180º

G Aq

q

q

rad 1º rad

3960 miles 13º 4 '180º 60 ' 180º

903.10miles

s r q

qG = 48º 9’

Glasgow

Albuquerque


4: Area of a Sector


1 1

In general:

A

A

q

q

For q1 = 2 ⇒ A1 = r 2

221

1

1

2 2

A rA r

q q q

q

The area A of the sector of a circle of radius r formed by a central

angle q radians is:

21

2A r q


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.






2

2

2

2

1

2

15ft 40º

2 180º

25ft

9

8.73 ft

A r q

Method 1: Recipe




Method 2: Conceptual



2

2

2

2

40

360

15ft

9

25ft

9

8.73 ft

A r


5: Linear & Angular Speed for Circular Motion


position at

time: t = 0

position at

time: t > 0

q

s v t

linear speedv

s

t

angular speed

t

q

v r


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.

rev rad2[ft] 180 2

min rev

ft720

min

ft2261.95

min

v r

r

v r



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 331/3

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.




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 331/3 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.

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 331/3 revolutions per minute. Present the

analysis to determine:




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 minute6

39

1

2

4

57

8

10

1112




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


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 3033.5 miles.

Therefore, a location on Earth at 40° north latitude is spinning on a

circle of radius 3033.5 miles. Compute the linear speed on the

surface of Earth at 40° north latitude.

distance travelled

elapsed time

2

2 3033.5miles

24hour

794.17 miles/hour

sv

t

r

T



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.

Documents

§5.1 Angles and Their Measure Objectives 1. Convert ...spot.pcc.edu/~kkidoguc/m112/m112_c05.1.pdf · Find the length of an arc of a circle. 3. ... band gare said to be co-terminal!