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THE UNIVERSITY OF AKRONTheoretical and Applied Mathematics
Flash CardsIntroduction to Angles
Katie Jonesand
Tom Price
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c© 2003 teprice@uakron.eduLast Revision Date: April 10, 2003 Version 1.0
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Convert 176 ◦ to radian measure.
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Convert 423 ◦ to radian measure.
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Convert 394 ◦ to radian measure.
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Convert5π8
radians to degrees.
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Convert3π11
radians to degrees.
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Convert 1.9024 radians to de-grees.
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Convert −3.45 radians to de-grees.
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Determine an angle of degreemeasure between 0 ◦ and 360 ◦
that is coterminal with an angleof measure 795 ◦ .
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Determine an angle of degreemeasure between 0 ◦ and 360 ◦
that is coterminal with an angleof measure −165 ◦ .
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Determine an angle of radianmeasure between 0 rad and2π rad that is coterminal with anangle of measure
23π3
rad .
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Are the angles 7π12 radians and
−19π12 radians coterminal?
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Are the angles −190 ◦ and 170 ◦
coterminal?
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Give the smallest positive andlargest negative angles which arecoterminal with but not equal toπ3 rad.
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Are the angles π12 radians and 15 ◦
coterminal?
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Draw a graph of the angle −495 ◦
using an arrow to indicate direc-tion.
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Draw a graph of the angle 5π3 rad .
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What is the radian measure of anangle that subtends a 14 unit arcon a circle of radius 5?
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What is the radian measure of anangle that subtends a 6 unit arcon a circle of radius 3?
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Determine the length of an arcsubtended by an angle measuring1.15 radians if the circle has a ra-dius of 149 units.
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Determine the length of an arcon a circle of radius 1
6 which issubtended by an angle of measure0.21 rad.
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To convert 176 ◦ to radian measure recallthat 180 ◦ describes an angle of measure πradians.
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Answer: t = 4445π rad
Solution: To convert 176 ◦ to radian measure we use the con-version formula
t
π=
θ
180.
We havet
π=
176180
=⇒ t =176π
180=
4445
π rad
or t = 3.0718 rad
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To convert 423 ◦ to radian measure recallthat 360 ◦ describes an angle of measure 2πradians.
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Answer: t = 423180π rad
Solution: To convert 423 ◦ to radian measure we use the con-version formula
t
π=
θ
180.
We havet
π=
423180
=⇒ t =423180
π rad
or t = 7.3827 rad
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To convert 394 ◦ to radian measure use theconversion formula.
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Answer: t = 19790 π
Solution: To convert 394 ◦ to radian measure we use the con-version formula
t
π=
θ
180.
We havet
π=
394180
=⇒ t =394180
π rad =19790
π rad
or t = 6. 876 6 rad
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HINT
To convert5π8
rad to degree measure use theconversion formula
t
π=
θ
180.
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Answer: θ = 112.5 ◦
Solution: To convert5π
8rad to degree measure we use the
conversion formulat
π=
θ
180.
We have5π8
π=
θ
180
=⇒ θ = 180 · 58
=225 ◦
2=⇒ θ = 112.5 ◦
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HINT
To convert3π11
rad to degree measure recallthat 180 ◦ describes an angle of measure πradians.
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Answer: θ = 49.091 ◦
Solution: To convert3π
11rad to degree measure we use the
conversion formulat
π=
θ
180.
We have3π11
π=
θ
180
=⇒ θ = 180 · 311
=540 ◦
11
=⇒ θ = 49.091 ◦
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HINT
To convert 1.9024 rad to degree measure re-call that 360 ◦ describes an angle of measure2π radians.
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Answer: θ = 109 ◦
Solution: To convert 1.9024 rad to degree measure we use theconversion formula
t
π=
θ
180.
We have1.9024
π=
θ
180
=⇒ θ = 180 · 1.9024π
=⇒ θ = 109 ◦
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HINT
To convert −3.45 rad to degree measure usethe conversion formula
t
π=
θ
180.
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Answer: θ = −197.67 ◦
Solution: To convert −3.45 rad to degree measure we use theconversion formula
t
π=
θ
180.
We have−3.45
π=
θ
180
=⇒ θ = 180 · −3.45π
=⇒ θ = −197. 67 ◦
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To find the angle coterminal with an anglemeasuring 795 ◦ recall that there are 360 ◦ inone full revolution of a circle.
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Answer: θ = 75 ◦
Solution: Determine how many full revolutions of the unitcircle there are in 795 ◦ by dividing 795 ◦ by 360 ◦.
795360
= 2 +75360
.
The quotient of 2 means that the given angle completes tworevolutions of the unit circle while the remainder of 75 in-dicates that the given angle is coterminal to one of measure75 ◦ . �
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To find the angle coterminal with an anglemeasuring −165 ◦ recall that negative anglesare measured in a clockwise direction.
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Answer: θ = 195 ◦
Solution: Subtract 165 ◦ from 360 ◦ to find the measure thatwill complete one revolution of a circle.
360 − 165 = 195 ◦
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To find the angle coterminal with an angle
measuring23π3
rad recall that there are 2πradians in one full revolution of a circle.
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Answer: t = 53π rad
Solution: Determine how many full revolutions of the unit
circle there are in23π
3rad by dividing
23π
3rad by 2π rad .
23π
32π
=236
= 3 +56.
The quotient of 3 means that the given angle completes threerevolutions of the unit circle (3 · 2π = 6π rad), while the re-mainder of 5
6 indicates that the given angle is coterminal toone of measure 5
6 · 2π = 53π rad, since 5
6 of a full revolution otthe unit circle is left over. �
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HINT
To determine if the angles 7π12 rad and
−19π12 rad are coterminal recall that cotermi-
nal angles have the same terminal sides.
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Answer: NoSolution: The difference of the two angles
7π
12− −19π
12=
136
π
is not an integer multiple of 2π, so the angles are NOT coter-minal.
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To determine if the angles −190 ◦ and 170 ◦
are coterminal recall that coterminal angleshave the same terminal sides.
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Answer: YesSolution: The difference of the two angles
−190 − 170 = −360
is an integer multiple of 360 ◦ , so the angles are coterminal.�
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HINT
To determine the smallest positive andlargest negative angles coterminal with butnot equal to π
3 rad, remember that to havethe same terminal side angles must have adifference that is an integer multiple of 2π.
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Answer: 7π3 rad and −5π
3 rad
Solution: Add 2π rad to π3 rad for the positive angle.
2π +π
3=
7π
3rad
Subtract 2π rad from π3 rad for the negative angle.
π
3− 2π = −5π
3rad
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To determine if the angles π12 rad and 15 ◦
are coterminal they must be measured in thesame units.
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Answer: YesSolution: Convert π
12 radians to degree measure using theconversion formula.
π12
π=
θ
180
=⇒ θ = 180 · 112
=⇒ θ = 15 ◦
The difference of the two angles
15 − 15 = 0.
is an integer multiple of 360 ◦ , so the angles are coterminal.Or without any calculations, an angle is always coterminal toitself, so 15 ◦ is coterminal with 15 ◦. �
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To draw a graph of the angle −495 ◦ remem-ber negative angles are measured in a clock-wise direction.
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Answer:
�
�
� � �� �
Solution: Starting at 0 ◦ and moving clockwise, make one fullrevolution then continue for another 135 ◦ because 135 = 495−360. �
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HINT
To draw a graph of the angle 5π3 rad divide
π rad into three equal parts.
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Answer:
�
�
Solution: Starting at 0 rad, move counter-clockwise by thirdsof π rad. Note, π rad = 3π
3 rad which is half of a circle. �
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To determine the angle that subtends a 14unit arc on a circle of radius 5, recall that tradians subtends an arc of length t · r, on acircle of radius r.
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Answer: t = 2.8 radSolution: The length, L, of a subtended arc is equal to theangle’s radian measure multiplied by the radius of the circle,giving the equation
L = t · r.
We have,
14 = t · 5
=⇒ t =145
= 2.8 rad
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To determine the angle that subtends a 6unit arc on a circle of radius 3, recall thatt radians subtends an arc of length t · r, on acircle of radius r.
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Answer: 2 radSolution: The length, L, of a subtended arc is equal to theangle’s radian measure multiplied by the radius of the circle,giving the equation
L = t · r.
We have,
6 = t · 3=⇒ t = 2 rad
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To find the length of an arc subtended byan angle measuring 1.15 rad on a circle witha radius of 149 units, recall that arc lengthis determined by multiplying the angle sub-tending it with the radius of the circle.
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Answer: L = 171. 35 unitsSolution: The length of a subtended arc, L, is equal to theangle’s radian measure multiplied by the radius of the circle,giving the equation
L = t · r.
We have,
L = 1.15 · 149= 171. 35 units
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To find the length of an arc on a circle of ra-dius 1
6 which is subtended by an angle mea-suring 0.21 rad, recall that arc length is de-termined by multiplying the angle subtend-ing it by the radius of the circle.
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Answer: L = 0.035 unitsSolution: The length of a subtended arc, L, is equal to theangle’s radian measure multiplied by the radius of the circle,giving the equation
L = t · r.
We have,
L = 0.21 · 16
= 0.035 units
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