Download Angles in a Unit Circle

Angles in a Unit Circle
by CHED on May 09, 2017
lesson duration of 6 minutes
under Precalculus
generated on May 09, 2017 at 06:47 am
Tags: Trigonometry
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Generated: May 09,2017 02:47 PM
Angles in a Unit Circle
( 6 mins )
Written By: CHED on July 3, 2016
Subjects: Precalculus
Tags: Trigonometry
Resources
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Content Standard
Key concepts of circular functions, trigonometric identities, inverse trigonometric functions, and the polar coordinate
system
Performance Standard
Formulate and solve accurately situational problems involving circular functions
Apply appropriate trigonometric identities in solving situational problems
Formulate and solve accurately situational problems involving appropriate trigonometric functions
Formulate and solve accurately situational problems involving the polar coordinate system
Learning Competencies
Illustrate the unit circle and the relationship between the linear and angular measures of a central angle in a unit circle
Introduction 1 mins
There are many problems involving angles in several fields like engineering, medical imaging, electronics, astronomy,
geography and many more. Surveyors, pilots, landscapers, designers, soldiers, and people in many other professions
heavily use angles and trigonometry to accomplish a variety of practical tasks. In this lesson, we will deal with the
basics of angle measures together with arc length and sectors.
Angle Measure 1 mins
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An angle is formed by rotating a ray about its endpoint. In the figure shown below, the initial side of angle /AOB is OA,
OA,
An angle
angle is
is said
said to
to be
be positive ifif the
the ray
ray rotates
rotates inin aa counterclockwise
counterclockwise direction,
direction, and
and the
the
while is terminal side is OB.
OB . An
angle is negative if it rotates in a clockwise direction.
Teaching Notes
Angles
in trigonometry
differ
angles
in Euclidean
geometry
in the
sense
of motion.
angle
in geometry
Angles
in trigonometry
differ
fromfrom
angles
in Euclidean
geometry
in the
sense
of motion.
An An
angle
in geometry
is is
defined
a union
of rays
is, static)
and measure
has measure
between
0 degrees
and degrees.
180 degrees.
An angle
defined
as a as
union
of rays
(that (that
is, static)
and has
between
0 degrees
and 180
An angle
in in
trigonometry is a rotation of a ray, and, therefore, has no limit. It has positive and negative directions and measures.
An angle
angle isis in
in standard position ififititisisdrawn
drawnininthe
the xy-plane
withitsitsvertex
vertexatatthe
theorigin
originand
anditsitsinitial
initialside
sideononthe
the
An
xy -planewith
positive x-axis. The angles a, B, and ? in the following figure are angles in standard position.
To measure angles, we use degrees, minutes, seconds, and radians.
For example, in degrees, minutes, and seconds,
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10 degrees 30’18” = 10(30+(18/60))
= 10 degrees 30.3’
= (10+(30.3/60) degrees
= 10.505 degrees
and
79.251 degrees = 79 degrees (0.251 x 60)’
= 79 degrees15.06’
= 79 degrees 15’(0.06 x 60)”
= 79 degrees 15’3.6”.
Recall that the unit circle is the circle with center at the origin and radius 1 unit.
trigonometry,as
asititwas
wasstudied
studiedininGrade
Grade9,9,the
thedegree
degreemeasure
measureisisoften
oftenused.
used.On
Onthe
theother
otherhand,
hand,ininsome
somefields
fieldsofof
InIntrigonometry,
mathematics like
like calculus,
calculus, radian
radian measure
measure of
of angles
angles is
is preferred.
preferred. Radian
Radian measure
measure allows
allows us
us to
to treat
treat the
the trigonometric
trigonometric
mathematics
functions as functions with the set of real numbers as domains, rather than angles.
Example 3.1.1. In the following figure, identify the terminal side of an angle in standard position with given measure.
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Since aa unit
unit circle
circle has
has circumference
circumference 22pi
pi, ,aacentral
centralangle
anglethat
thatmeasures
measures360
360degrees
degreeshas
hasmeasure
measureequivalent
equivalenttoto22pi
pi
Since
radians. Thus, we obtain the following conversion rules.
Figure3.2
3.2shows
showssome
somespecial
specialangles
anglesininstandard
standardposition
positionwith
withthe
theindicated
indicatedterminal
terminalsides.
sides.The
Thedegree
degreeand
andradian
radian
Figure
measures are also given.
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Example 3.1.2. Express 75 degrees and 240 degrees in radians.
Solution.
75 (pi/180) = 5pi/12 => 75 degrees = 5pi/12 rad
240 (pi/180) = 4pi/3 => 240 degrees = 4pi/3 rad
Example 3.1.3. Express pi/8 rad and 11pi/6 rad in degrees.
Solution.
pi/8 (180/pi) = 22.5 => pi/8 rad = 22.5 degrees
11pi/6 (180/pi) = 330 => 11pi/6 rad = 330 degrees
Seatwork 1 mins
Seatwork/Homework
1. Convert the following degree measures to radian measure.
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2. Convert the following radian measures to degree measure.
Coterminal Angles 1 mins
Two angles in standard position that have a common terminal side are called coterminal angles.
angles. Observe that the
degree measures of coterminal angles differ by multiples of 360 degrees.
As a quick illustration, to find one coterminal angle with an angle that measures 410 degrees, just subtract 360
degrees, resulting in 50 degrees. See Figure 3.3.
Example 3.1.4. Find the angle coterminal with ?380 degrees that has measure
(1) between 0 degrees and 360 degrees, and
(2) between ?360 degrees and 0 degrees.
Solution. A negative angle moves in a clockwise direction, and the angle ?380 degrees lies in Quadrant IV.
(1) ?380 degrees + 2 360 degrees = 340 degrees
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(2) ?380 degrees + 360 degrees = ?20 degrees
Seatwork/Homework
1. Find the angle between 0 degrees and 360 degrees (if in degrees) or between 0 rad and 2pi rad (if in radians) that is
coterminal with the given angle.
2. Find the angle between ?360 degrees and 0 degrees (if in degrees) or between ?2pi rad and 0 rad (if in radians)
that is coterminal with the given angle.
Seatwork 0 mins
Seatwork/Homework
1. Find the angle between 0 degrees and 360 degrees (if in degrees) or between 0 rad and 2pi rad (if in radians) that is
coterminal with the given angle.
2. Find the angle between ?360 degrees and 0 degrees (if in degrees) or between ?2pi rad and 0 rad (if in radians)
that is coterminal with the given angle.
Arc Length and Area of a Sector 1 mins
In a circle, a central angle whose radian measure is ? subtends an arc that is the fraction ?/2pi of the circumference of
the circle. Thus, in a circle of radius r (see Figure 3.4), the length s of an arc that subtends the angle ? is
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Teaching Notes
Review how arcs were measured in Grade 10. What unit of measure was used pi for two circles with different radii, do
equalcentral
centralangles
anglesintercept
interceptarcs
arcsofofthe
thesame
samemeasure
measurepi.pi.Conclude
Concludethat
thatprevious
previousnotion
notionofofarc
arcmeasure
measureisisnot
notthe
the
equal
same as length. Arcs are now measured in terms of length and measure changes with the radius of the circle.
Example 3.1.5. Find the length of an arc of a circle with radius 10 m that subtends a central angle of 30 degrees.
Solution. Since
Since
given
central
angle
in degrees,
have
to convert
it into
radian
measure.
Then
apply
the the
given
central
angle
is inisdegrees,
we we
have
to convert
it into
radian
measure.
Then
apply
the the
formula for an arc length.
Example 3.1.6. A
A central
central angle
angle ?? in
in aa circle
circle of
of radius
radius 44 m
m is
is subtended
subtended by
by an
an arc
arc of
of length
length 66 m.
m. Find
Find the
the measure
measure of
of ??
in radians.
Solution.
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? = s/r = 6/4 = 3/2rad
sectorofofa acircle
circleisisthe
theportion
portionofofthe
theinterior
interiorofofa acircle
circlebounded
boundedbybythe
theinitial
initialand
andterminal
terminalsides
sidesofofa acentral
centralangle
angle
AAsector
and its
its intercepted
intercepted arc.
arc. ItIt is
is like
like aa “slice
“slice of
of pizza.”
pizza.” Note
Note that
that an
an angle
angle with
with measure
measure 2pi
2pi radians
radians will
will define
define aa sector
sector that
that
and
corresponds
to the
whole
“pizza.”
Therefore,
a central
angle
of sector
a sector
measure
? radians,
then
sector
corresponds
to the
whole
“pizza.”
Therefore,
if aif central
angle
of a
hashas
measure
? radians,
then
thethe
sector
makesupupthe
thefraction
fraction?/2pi
?/2piofofa acomplete
completecircle.
circle.See
SeeFigure
Figure3.5.
3.5.Since
Sincethe
thearea
areaofofa acomplete
completecircle
circlewith
withradius
radius r is
makes
pir
pir^2, we have
Example 3.1.7. Find the area of a sector of a circle with central angle 60 degrees if the radius of the circle is 3 m.
Solution. First, we have to convert 60 degrees into radians. Then apply the formula for computing the area of a sector.
Example 3.1.8. A sprinkler on a golf course fairway is set to spray water over a distance of 70 feet and rotates through
an angle of 120 degrees. Find the area of the fairway watered by the sprinkler.
Solution.
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Seatwork/Homework
1. In a circle of radius 7 feet, find the length of the arc that subtends a central angle of 5 radians.
Answer: 35 ft
2. A central angle pi in a circle of radius 20 m is subtended by an arc of length 15pi m. Find the measure of ? in
degrees.
Answer: 135 degrees
3. Find the area of a sector of a circle with central angle that measures 75 degrees if the radius of the circle is 6 m.
Answer: 7.5 m^2
Exercises 3.1
1. Give the degree/radian measure of the following special angles.
2. Convert each degree measure to radians. Leave answers in terms of pi.
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3. Convert each radian measure to degree-minute-second measure (approximate if necessary).
4. Find the angle between 0 degrees and 360 degrees (if in degrees) or between 0 rad and 2pi rad (if in radians) that is
coterminal with the given angle.
5. Find the angle between -360 degrees and 0 degrees (if in degrees) or between ?2pi rad and 0 rad (if in radians) that
is coterminal with the given angle.
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6. Find the length of an arc of a circle with radius 21 m that subtends a central angle of 15 degrees.
Answer: 7pi/4 m
7. A central angle ? in a circle of radius 9 m is subtended by an arc of length 12 m. Find the measure of ? in radians.
Answer: 4/3 rad
8. Find the radius of a circle in which a central angle of pi/6 rad determines a sector of area 64 m^2.
Answer: 16 m
9. If the radius of a circle is doubled, how is the length of the arc intercepted by a fixed central angle changed?
Answer: The length is doubled.
10. Radian measure simplifies many formulas, such as the formula for arc length, s = r ?. Give the corresponding
formula when ? is measured in degrees instead of radians.
Answer: s = pir
pir ?/180
11. As shown below, find the radius of the pulley if a rotation of 51.6 degrees raises the weight by 11.4 cm.
Answer: 12.7 cm
12. How many inches will the weight rise if the pulley whose radius is 9.27 inches is rotated through an angle of 71
degrees 500pi
Answer: 11.6 in
13. Continuing with the previous item, through what angle (to the nearest minute) must the pulley be rotated to raise
the weight 6 in?
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Answer: 37 degrees 5'
14. Given a circle of radius 3 in, find the measure (in radians) of the central angle of a sector of area 16 in^2.
Answer: 3.6 rad
15. An automatic lawn sprinkler sprays up to a distance of 20 feet while rotating 30 degrees. What is the area of the
sector the sprinkler covers?
Answer: 104.72 ft^2
16. A jeepney has a windshield wiper on the driver’s side that has total arm and blade 10 inches long and rotates back
and forth through an angle of 95 degrees. The shaded region in the figure is the portion of the windshield cleaned by
the 7-inch wiper blade. What is the area of the region cleaned?
Answer: 75.4 in^2
17. If the radius of a circle is doubled and the central angle of a sector is unchanged, how is the area of the sector
changed?
Answer: The area is quadrupled.
18.Give the corresponding formula for the area of a sector when the angle is measured in degrees.
Answer: A = pir2? 360
19.A frequent problem in surveying city lots and rural lands adjacent to curves of highways and railways is that of
finding the area when one or more of the boundary lines is the arc of a circle. Approximate the total area of the lot
shown in the figure.
Answer: 1909.0 m^2
20. Two gears of radii 2.5 cm and 4.8 cm are adjusted so that the smaller gear drives the larger one, as shown. If the
smaller gear rotates counterclockwise through 225 degrees, through how many degrees will the larger gear rotate?
Answer: 117 degrees
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Seatwork 1 mins
Seatwork/Homework
1. In a circle of radius 7 feet, find the length of the arc that subtends a central angle of 5 radians.
Answer: 35 ft
2. A central angle pi in a circle of radius 20 m is subtended by an arc of length 15pi m. Find the measure of ? in
degrees.
Answer: 135 degrees
3. Find the area of a sector of a circle with central angle that measures 75 degrees if the radius of the circle is 6 m.
Answer: 7.5 m^2
Exercises 3.1
1. Give the degree/radian measure of the following special angles.
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2. Convert each degree measure to radians. Leave answers in terms of pi.
3. Convert each radian measure to degree-minute-second measure (approximate if necessary).
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4. Find the angle between 0 degrees and 360 degrees (if in degrees) or between 0 rad and 2pi rad (if in radians) that is
coterminal with the given angle.
5. Find the angle between -360 degrees and 0 degrees (if in degrees) or between ?2pi rad and 0 rad (if in radians) that
is coterminal with the given angle.
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6. Find the length of an arc of a circle with radius 21 m that subtends a central angle of 15 degrees.
Answer: 7pi/4 m
7. A central angle ? in a circle of radius 9 m is subtended by an arc of length 12 m. Find the measure of ? in radians.
Answer: 4/3 rad
8. Find the radius of a circle in which a central angle of pi/6 rad determines a sector of area 64 m^2.
Answer: 16 m
9. If the radius of a circle is doubled, how is the length of the arc intercepted by a fixed central angle changed?
Answer: The length is doubled.
10. Radian measure simplifies many formulas, such as the formula for arc length, s = r ?. Give the corresponding
formula when ? is measured in degrees instead of radians.
Answer: s = pir
pir ?/180
11. As shown below, find the radius of the pulley if a rotation of 51.6 degrees raises the weight by 11.4 cm.
Answer: 12.7 cm
12. How many inches will the weight rise if the pulley whose radius is 9.27 inches is rotated through an angle of 71
degrees 500pi
Answer: 11.6 in
13. Continuing with the previous item, through what angle (to the nearest minute) must the pulley be rotated to raise
the weight 6 in?
Answer: 37 degrees 5'
14. Given a circle of radius 3 in, find the measure (in radians) of the central angle of a sector of area 16 in^2.
Answer: 3.6 rad
15. An automatic lawn sprinkler sprays up to a distance of 20 feet while rotating 30 degrees. What is the area of the
sector the sprinkler covers?
Answer: 104.72 ft^2
16. A jeepney has a windshield wiper on the driver’s side that has total arm and blade 10 inches long and rotates back
and forth through an angle of 95 degrees. The shaded region in the figure is the portion of the windshield cleaned by
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the 7-inch wiper blade. What is the area of the region cleaned?
Answer: 75.4 in^2
17. If the radius of a circle is doubled and the central angle of a sector is unchanged, how is the area of the sector
changed?
Answer: The area is quadrupled.
18.Give the corresponding formula for the area of a sector when the angle is measured in degrees.
Answer: A = pir2? 360
19.A frequent problem in surveying city lots and rural lands adjacent to curves of highways and railways is that of
finding the area when one or more of the boundary lines is the arc of a circle. Approximate the total area of the lot
shown in the figure.
Answer: 1909.0 m^2
20. Two gears of radii 2.5 cm and 4.8 cm are adjusted so that the smaller gear drives the larger one, as shown. If the
smaller gear rotates counterclockwise through 225 degrees, through how many degrees will the larger gear rotate?
Answer: 117 degrees
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