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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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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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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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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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HINT

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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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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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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