math hands
TRIGONOMETRY CH 01 SEC 01 NOTES
math hands
ANGLES intro
The Main Idea In this section we discuss important ideas and vocabulary associated with angles. We will discuss the two most common units of measurement for angles, degrees and radians. We will measure famous angles using both units, learn how to convert measurements from radians to degrees and vise-versa. We also introduce timeless ideas such as the concept of π. Units of Measurement Let us be clear, one central idea in trig is to be able to measure distances far away, such as the hight of a building, for measuring angles close by, such as the angle one makes by considering the lines of sight from the top to the building and to the bottom of the building. The underlying reason for the interest in this is of course because small angles make small opposite sides while larger angles make larger opposite sides. Dominating this relationship implies being able to take measurements of things from very far away, and this is precisely the essence of trigonometry. This is precisely what we will do in this course. Moreover, to carry out this amazing feat we must become good at measuring angles. ”Little angle”, ”medium” or ”really big angle” are descriptive terms but we can do much better. Indeed, we do much better by developing ”units of measurement” for angles. We will adopt two such units, degrees and radians. We will define and explain exactly what each one of these units represents. We will then consider famous angles and their respective famous measurements both in radians and degrees. We will then conclude with take apart and digest the timeless concept of π ( pronounced ’pi’). Degrees Imagine taking a circle, then slicing it up in half, then in quarters, etc... until there are 360 slices of equal size. One Degree is by definition the measurement of the angle for each of these slices. It may be interesting to note that there 360 is in some ways an arbitrary number which well cold have been 100 or 1000. For whatever, reason 360 was the number of slices needed to make each one ”one degree”. These ”degrees” have remained popular since inception and are now part of our everyday language (ie we make an 180 turn.. etc.). Let us consider some famous angles measured in degrees. Degrees are units of measurement for angles. One degree is defined to be the measurement of the angle created by slicing up a circle into 360 equal slices. One Degree angle 1 360
of a circle
Radians Radians are in some ways more natural than degrees. To understand what a radian is, we take a look at how one such radian is ’cooked’. To do so, we begin with a circle with radius r, where r is some positive real number. We then imagine measuring this same distance r, along a portion of the circle, marking the endpoints. From each of these endpoints, we draw a segment to the center of the circle, constructing a very important slice of the circle. Such slice should look something like this.
pg. 4
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2007-2011 MathHands.com v
TRIGONOMETRY CH 01 SEC 01 NOTES
ANGLES intro
math hands An important slice of a circle, carefully constructed
r
r The angle created by this slice is precisely what we will define as the measurement of one radian. Said differently, one radian is defined to be the measurement of the angle created by the stretching along a circle the length of one radius. Radians are units of measurement for angles. One Radian is defined to be the measurement of the angle created by the distance of one radius along part of the circumference of a circle. One Radian
r
r
This discussion turns into some timeless ideas as we ask some innocent little questions. For example, we noted before the number of ”one-degree slices” that fit in a circle is 360, by definition. Thus a half-circle contains exactly 180 one-degree slices, or said differently, the half-circle angle measures 180◦. We now ask the same question for radians: how many one-radian slices fit in the half-circle?. This undecorated question leads to big, timeless, and very decorated ideas. As it turns out we can sketch these slices and see that roughly 3 such slices fit in the half-circle. But if we are extremely careful and accurate in our drawings we realize 3 such slices fall short of the half-circle, thus we can conclude that a little more than 3 radians fit in the half-circle. We can take portions of a radius and fit them in to conclude that 3.1 radians fit in a half-circle, but actually there is still room left for more. Depending on our instruments we may be able to realize that 3.14 radians fit in a half-circle, but there is a little more room left. The question of exactly how many radians fit in a half-circle consumed bright minds and ancient civilizations for centuries. Some civilizations never got to a satisfactory answer to this while others answered it incorrectly. Using some very deep ideas beyond the scope of this course, we now know the number starts off as π = 3.14159265358979323846264338327950288 . . . The trouble is that is does not end there, and to stop there would be inaccurate. In fact, it goes on for millions and millions and more digits. The contemporarily prevailing resolution has been to recognize that this number can not be named by its digits-representation due to the fact that it has too many digits without much of a pattern. Instead, we name this number and refer to it by the symbol ”π”.
pg. 5
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2007-2011 MathHands.com v.1010
TRIGONOMETRY
Definition of
ANGLES intro
math hands
CH 01 SEC 01 NOTES
π
Thus, the number that can not be named by digits, π, was born. It was born to be the exact number radians that fit in a half-circle. We emphasize: π is defined to be the exact number of times the radius of a circle fits in half the circumference of the circle. OWN π r r
r
π = exact number of r’s that fit in a half-circle π ≈ 3.14
Famous Angles: Degrees & Radians The above definitions of a degree, a radian, and π lead to some famous consequences. For example, if by definition, there are 360◦ in a circle, then there are 180◦ in a half-circle, and 90◦ in a quarter circle. By similar logic, there are π radians on a half-circle, 2π in a complete circle, and π2 radians in a quarter of a circle. This leads to famous conversions, π radians can be converted to 180◦ just as 2πrad = 360◦ , or π2 = 90◦ . It should be noted that when an angle measurement is given with no specified units, the default units are by default and tradition radians. Here is a visual representation of some famous angles converted expressed in both degrees and in radians. 7π 6
π = 180◦
π 2
= 90◦
π 3
= 60◦
= 7 · 30◦ = 210◦
= 30◦
π 6
Famous: Degrees & Radians Here is another way to visualize the conversion of famous degrees and their respective famous counterparts measured in radians. 0◦
180◦ ◦
0
◦
◦
◦
◦
30
60
90
120
150
π 6
2π 6
π 2
4π 6
5π 6
360◦ ◦
π
◦
◦
◦
◦
240
210
270
300
330
8π 6
7π 6
3π 2
10π 6
11π 6
2π
Now we convert a non-famous angle. The key idea to convert units from/to degrees to/from radians is the observation, by definition of degrees and radians [DDR]:
pg. 6
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2007-2011 MathHands.com v.1010
TRIGONOMETRY
ANGLES intro
math hands
CH 01 SEC 01 NOTES
π rad = 180 deg thus.. 1 =
π rad 180 deg
and 1 =
180 deg π rad
Converting can be achieved by multiplying by the appropriate versions of ’1’. To convert from degrees to radians π rad one would multiply by ’1’ written as 1 = 180 deg . On the other hand, to convert from radians to degrees one would multiply by ’1’ written as 1 =
180 deg π rad
example: Convert 75 deg 75 deg = 75 deg · 1
(MiD)
π rad 180 deg π rad = 75 · 180 ≈ 1.309 rad
= 75 deg ·
(DDR) (Bi, note ’deg’ no more) (Bi)
Interior Angels Theorem The following theorem will be used frequently in our course. It states that in each of the triangles we will study, the sum of the interior angles is exactly 180◦ . It should be noted that this theorem works well for triangles on flat surfaces and may not hold true for non-flat surfaces such as the surface of a sphere. Moreover, the student is encouraged to explain or attempt to explain why it is so.
For any triangle in Euclidian Space, the sum of the interior angles is 180◦ i.e. for α, β and σ below σ α
e con-
β then
α + β + σ = 180◦
clude this section with a couple words to add to our vocabulary. If the measurement of two angles adds to 90◦ , these angles are called complimentary Note: on a right triangle, the two non-right angles are complimentary. i.e. α and β below α + β = 90◦
β
α If the measurement of two angles adds to 180◦, these angles are called supplementary. Note: on a right triangle, the two non-right angles are complimentary. i.e. α and β below
α + β = 180◦ α
pg. 7
β
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2007-2011 MathHands.com v.1010
TRIGONOMETRY CH 01 SEC 01 HW WORKSHEET
1. Convert 300
2. Convert 30
ANGLES intro
math hands
◦
◦
7. Convert 3
◦
8. Convert 1
◦
9. Convert
−5π 6
3. How many radii fit on the circumference of half a circle?
10. What is one radian?
4. How many radii fit on an entire circle
11. Convert 170
◦
5. If the radius of a circle is 5 inches long, how long is the circumference of half of the circle? 12. If the radius of a circle is 7 inches long, how long is the circumference of the entire circle.
6. Convert 1 13. Convert 4
pg. 1
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2007-2011 MathHands.com v.1010
TRIGONOMETRY CH 01 SEC 01 HW WORKSHEET
20. If the radius of a circle is 5 inches long, how long is the circumference of the circle?
14. Convert 3
15. Convert
ANGLES intro
math hands
5π 12
21. Determine if: π = 3.14 A. TRUE
16. Convert 50
B. FALSE
◦
22. Convert 1 rad
17. Determine if: π ≈ 3.14 A. TRUE
18. Convert 60
B. FALSE
◦
19. Convert 225
pg. 2
23. If the radius of a circle is blah inches long, how long is the circumference of the entire circle.
24. Convert
◦
11π 12
25. Convert 390
◦
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2007-2011 MathHands.com v.1010
TRIGONOMETRY CH 01 SEC 01 HW WORKSHEET
ANGLES intro
math hands
26. Convert 2 29. What is one degree?
27. Convert
π 12
30. Convert 10
◦
28. If the radius of a circle is r inches long, how long is the circumference of the entire circle.
pg. 3
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2007-2011 MathHands.com v.1010
TRIGONOMETRY
◦
◦
1. Convert 300
7. Convert 3
π rad 180
13. Convert 4
Solution:
Solution: 300 ·
ANGLES intro
math hands
CH 01 SEC 01 SOLUTIONS
=
◦
3 ·
5π 3
πrad ◦ 180
=
π rad 60
Solution: 4·
◦
◦
180 πrad
≈ 229.18
◦
8. Convert 1
◦
2. Convert 30
Solution:
Solution:
◦
1 ·
π/6
πrad ◦ 180
=
14. Convert 3
π rad 180
Solution:
3. How many radii fit on the circumference of half a circle?
9. Convert
3·
−5π 6
◦
◦
180 πrad
≈ 171.89
Solution: ◦
Solution: π radii fit on the circumference of half a circle.
each π/6 is 30 thus... −5π 6
◦
◦
= −5(30 ) = −150
15. Convert
5π 12
Solution: 4. How many radii fit on an entire circle
Solution: 2π radii fit on the circumference of a circle.
5. If the radius of a circle is 5 inches long, how long is the circumference of half of the circle?
Solution:
10. What is one radian? Solution: unit of measurement of angles; equivalent to the measurement of the angles created by an arc of a circle of the same size as the radius of the circle.
5π 12
◦
·
◦
180 π
= 75
◦
16. Convert 50
Solution: ◦
50 ·
πrad ◦ 180
=
5π 18
◦
11. Convert 170
Solution:
π · 5 ≈ 3.1416 × 5 = 15.708
◦
170 ·
πrad ◦ 180
=
17π 18
17. Determine if: π ≈ 3.14 A. True
6. Convert 1
12. If the radius of a circle is 7 inches long, how long is the circumference of the entire circle.
Solution: 1·
◦
180 π
B. False
◦
≈ 57.2956
Solution:
◦
18. Convert 60
2·π·7
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2007-2011 MathHands.com v.1010
TRIGONOMETRY
ANGLES intro
math hands
CH 01 SEC 01 SOLUTIONS
27. Convert
π 12
Solution: 23. If the radius of a circle is blah inches long, how long is the circumference of the entire circle.
π/3
◦
19. Convert 225
Solution: π 12
◦
·
◦
180 π
= 15
Solution: 2 · π · blah
Solution: 225 ·
π rad 180
=
28. If the radius of a circle is r inches long, how long is the circumference of the entire circle.
5π 4
24. Convert
11π 12
Solution: 20. If the radius of a circle is 5 inches long, how long is the circumference of the circle?
2·π·r
Solution: 11π 12
◦
·
◦
180 π
= 165
Solution: 29. What is one degree?
2·π·5 ◦
25. Convert 390
Solution:
21. Determine if:
390 ·
π rad 180
=
13π 6
Solution: unit of measurement of angles; equivalent to the measurement of the angles created by 1/360 of a circle.
π = 3.14 A. True B. False ◦
30. Convert 10
26. Convert 2 22. Convert 1 rad
2·
Solution: 1·
pg. 2
◦
180 πrad
Solution:
Solution: ◦
180 πrad
◦
≈ 114.59
10 ·
π rad 180
=
π rad 18
≈ .175rad
◦
≈ 57.2956
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2007-2011 MathHands.com v.1010
math hands
ANGLES intro
The Main Idea In this section we discuss important ideas and vocabulary associated with angles. We will discuss the two most common units of measurement for angles, degrees and radians. We will measure famous angles using both units, learn how to convert measurements from radians to degrees and vise-versa. We also introduce timeless ideas such as the concept of π. Units of Measurement Let us be clear, one central idea in trig is to be able to measure distances far away, such as the hight of a building, for measuring angles close by, such as the angle one makes by considering the lines of sight from the top to the building and to the bottom of the building. The underlying reason for the interest in this is of course because small angles make small opposite sides while larger angles make larger opposite sides. Dominating this relationship implies being able to take measurements of things from very far away, and this is precisely the essence of trigonometry. This is precisely what we will do in this course. Moreover, to carry out this amazing feat we must become good at measuring angles. ”Little angle”, ”medium” or ”really big angle” are descriptive terms but we can do much better. Indeed, we do much better by developing ”units of measurement” for angles. We will adopt two such units, degrees and radians. We will define and explain exactly what each one of these units represents. We will then consider famous angles and their respective famous measurements both in radians and degrees. We will then conclude with take apart and digest the timeless concept of π ( pronounced ’pi’). Degrees Imagine taking a circle, then slicing it up in half, then in quarters, etc... until there are 360 slices of equal size. One Degree is by definition the measurement of the angle for each of these slices. It may be interesting to note that there 360 is in some ways an arbitrary number which well cold have been 100 or 1000. For whatever, reason 360 was the number of slices needed to make each one ”one degree”. These ”degrees” have remained popular since inception and are now part of our everyday language (ie we make an 180 turn.. etc.). Let us consider some famous angles measured in degrees. Degrees are units of measurement for angles. One degree is defined to be the measurement of the angle created by slicing up a circle into 360 equal slices. One Degree angle 1 360
of a circle
Radians Radians are in some ways more natural than degrees. To understand what a radian is, we take a look at how one such radian is ’cooked’. To do so, we begin with a circle with radius r, where r is some positive real number. We then imagine measuring this same distance r, along a portion of the circle, marking the endpoints. From each of these endpoints, we draw a segment to the center of the circle, constructing a very important slice of the circle. Such slice should look something like this.
pg. 4
c
2007-2011 MathHands.com v
TRIGONOMETRY CH 01 SEC 01 NOTES
ANGLES intro
math hands An important slice of a circle, carefully constructed
r
r The angle created by this slice is precisely what we will define as the measurement of one radian. Said differently, one radian is defined to be the measurement of the angle created by the stretching along a circle the length of one radius. Radians are units of measurement for angles. One Radian is defined to be the measurement of the angle created by the distance of one radius along part of the circumference of a circle. One Radian
r
r
This discussion turns into some timeless ideas as we ask some innocent little questions. For example, we noted before the number of ”one-degree slices” that fit in a circle is 360, by definition. Thus a half-circle contains exactly 180 one-degree slices, or said differently, the half-circle angle measures 180◦. We now ask the same question for radians: how many one-radian slices fit in the half-circle?. This undecorated question leads to big, timeless, and very decorated ideas. As it turns out we can sketch these slices and see that roughly 3 such slices fit in the half-circle. But if we are extremely careful and accurate in our drawings we realize 3 such slices fall short of the half-circle, thus we can conclude that a little more than 3 radians fit in the half-circle. We can take portions of a radius and fit them in to conclude that 3.1 radians fit in a half-circle, but actually there is still room left for more. Depending on our instruments we may be able to realize that 3.14 radians fit in a half-circle, but there is a little more room left. The question of exactly how many radians fit in a half-circle consumed bright minds and ancient civilizations for centuries. Some civilizations never got to a satisfactory answer to this while others answered it incorrectly. Using some very deep ideas beyond the scope of this course, we now know the number starts off as π = 3.14159265358979323846264338327950288 . . . The trouble is that is does not end there, and to stop there would be inaccurate. In fact, it goes on for millions and millions and more digits. The contemporarily prevailing resolution has been to recognize that this number can not be named by its digits-representation due to the fact that it has too many digits without much of a pattern. Instead, we name this number and refer to it by the symbol ”π”.
pg. 5
c
2007-2011 MathHands.com v.1010
TRIGONOMETRY
Definition of
ANGLES intro
math hands
CH 01 SEC 01 NOTES
π
Thus, the number that can not be named by digits, π, was born. It was born to be the exact number radians that fit in a half-circle. We emphasize: π is defined to be the exact number of times the radius of a circle fits in half the circumference of the circle. OWN π r r
r
π = exact number of r’s that fit in a half-circle π ≈ 3.14
Famous Angles: Degrees & Radians The above definitions of a degree, a radian, and π lead to some famous consequences. For example, if by definition, there are 360◦ in a circle, then there are 180◦ in a half-circle, and 90◦ in a quarter circle. By similar logic, there are π radians on a half-circle, 2π in a complete circle, and π2 radians in a quarter of a circle. This leads to famous conversions, π radians can be converted to 180◦ just as 2πrad = 360◦ , or π2 = 90◦ . It should be noted that when an angle measurement is given with no specified units, the default units are by default and tradition radians. Here is a visual representation of some famous angles converted expressed in both degrees and in radians. 7π 6
π = 180◦
π 2
= 90◦
π 3
= 60◦
= 7 · 30◦ = 210◦
= 30◦
π 6
Famous: Degrees & Radians Here is another way to visualize the conversion of famous degrees and their respective famous counterparts measured in radians. 0◦
180◦ ◦
0
◦
◦
◦
◦
30
60
90
120
150
π 6
2π 6
π 2
4π 6
5π 6
360◦ ◦
π
◦
◦
◦
◦
240
210
270
300
330
8π 6
7π 6
3π 2
10π 6
11π 6
2π
Now we convert a non-famous angle. The key idea to convert units from/to degrees to/from radians is the observation, by definition of degrees and radians [DDR]:
pg. 6
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2007-2011 MathHands.com v.1010
TRIGONOMETRY
ANGLES intro
math hands
CH 01 SEC 01 NOTES
π rad = 180 deg thus.. 1 =
π rad 180 deg
and 1 =
180 deg π rad
Converting can be achieved by multiplying by the appropriate versions of ’1’. To convert from degrees to radians π rad one would multiply by ’1’ written as 1 = 180 deg . On the other hand, to convert from radians to degrees one would multiply by ’1’ written as 1 =
180 deg π rad
example: Convert 75 deg 75 deg = 75 deg · 1
(MiD)
π rad 180 deg π rad = 75 · 180 ≈ 1.309 rad
= 75 deg ·
(DDR) (Bi, note ’deg’ no more) (Bi)
Interior Angels Theorem The following theorem will be used frequently in our course. It states that in each of the triangles we will study, the sum of the interior angles is exactly 180◦ . It should be noted that this theorem works well for triangles on flat surfaces and may not hold true for non-flat surfaces such as the surface of a sphere. Moreover, the student is encouraged to explain or attempt to explain why it is so.
For any triangle in Euclidian Space, the sum of the interior angles is 180◦ i.e. for α, β and σ below σ α
e con-
β then
α + β + σ = 180◦
clude this section with a couple words to add to our vocabulary. If the measurement of two angles adds to 90◦ , these angles are called complimentary Note: on a right triangle, the two non-right angles are complimentary. i.e. α and β below α + β = 90◦
β
α If the measurement of two angles adds to 180◦, these angles are called supplementary. Note: on a right triangle, the two non-right angles are complimentary. i.e. α and β below
α + β = 180◦ α
pg. 7
β
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2007-2011 MathHands.com v.1010
TRIGONOMETRY CH 01 SEC 01 HW WORKSHEET
1. Convert 300
2. Convert 30
ANGLES intro
math hands
◦
◦
7. Convert 3
◦
8. Convert 1
◦
9. Convert
−5π 6
3. How many radii fit on the circumference of half a circle?
10. What is one radian?
4. How many radii fit on an entire circle
11. Convert 170
◦
5. If the radius of a circle is 5 inches long, how long is the circumference of half of the circle? 12. If the radius of a circle is 7 inches long, how long is the circumference of the entire circle.
6. Convert 1 13. Convert 4
pg. 1
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2007-2011 MathHands.com v.1010
TRIGONOMETRY CH 01 SEC 01 HW WORKSHEET
20. If the radius of a circle is 5 inches long, how long is the circumference of the circle?
14. Convert 3
15. Convert
ANGLES intro
math hands
5π 12
21. Determine if: π = 3.14 A. TRUE
16. Convert 50
B. FALSE
◦
22. Convert 1 rad
17. Determine if: π ≈ 3.14 A. TRUE
18. Convert 60
B. FALSE
◦
19. Convert 225
pg. 2
23. If the radius of a circle is blah inches long, how long is the circumference of the entire circle.
24. Convert
◦
11π 12
25. Convert 390
◦
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2007-2011 MathHands.com v.1010
TRIGONOMETRY CH 01 SEC 01 HW WORKSHEET
ANGLES intro
math hands
26. Convert 2 29. What is one degree?
27. Convert
π 12
30. Convert 10
◦
28. If the radius of a circle is r inches long, how long is the circumference of the entire circle.
pg. 3
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2007-2011 MathHands.com v.1010
TRIGONOMETRY
◦
◦
1. Convert 300
7. Convert 3
π rad 180
13. Convert 4
Solution:
Solution: 300 ·
ANGLES intro
math hands
CH 01 SEC 01 SOLUTIONS
=
◦
3 ·
5π 3
πrad ◦ 180
=
π rad 60
Solution: 4·
◦
◦
180 πrad
≈ 229.18
◦
8. Convert 1
◦
2. Convert 30
Solution:
Solution:
◦
1 ·
π/6
πrad ◦ 180
=
14. Convert 3
π rad 180
Solution:
3. How many radii fit on the circumference of half a circle?
9. Convert
3·
−5π 6
◦
◦
180 πrad
≈ 171.89
Solution: ◦
Solution: π radii fit on the circumference of half a circle.
each π/6 is 30 thus... −5π 6
◦
◦
= −5(30 ) = −150
15. Convert
5π 12
Solution: 4. How many radii fit on an entire circle
Solution: 2π radii fit on the circumference of a circle.
5. If the radius of a circle is 5 inches long, how long is the circumference of half of the circle?
Solution:
10. What is one radian? Solution: unit of measurement of angles; equivalent to the measurement of the angles created by an arc of a circle of the same size as the radius of the circle.
5π 12
◦
·
◦
180 π
= 75
◦
16. Convert 50
Solution: ◦
50 ·
πrad ◦ 180
=
5π 18
◦
11. Convert 170
Solution:
π · 5 ≈ 3.1416 × 5 = 15.708
◦
170 ·
πrad ◦ 180
=
17π 18
17. Determine if: π ≈ 3.14 A. True
6. Convert 1
12. If the radius of a circle is 7 inches long, how long is the circumference of the entire circle.
Solution: 1·
◦
180 π
B. False
◦
≈ 57.2956
Solution:
◦
18. Convert 60
2·π·7
pg. 1
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2007-2011 MathHands.com v.1010
TRIGONOMETRY
ANGLES intro
math hands
CH 01 SEC 01 SOLUTIONS
27. Convert
π 12
Solution: 23. If the radius of a circle is blah inches long, how long is the circumference of the entire circle.
π/3
◦
19. Convert 225
Solution: π 12
◦
·
◦
180 π
= 15
Solution: 2 · π · blah
Solution: 225 ·
π rad 180
=
28. If the radius of a circle is r inches long, how long is the circumference of the entire circle.
5π 4
24. Convert
11π 12
Solution: 20. If the radius of a circle is 5 inches long, how long is the circumference of the circle?
2·π·r
Solution: 11π 12
◦
·
◦
180 π
= 165
Solution: 29. What is one degree?
2·π·5 ◦
25. Convert 390
Solution:
21. Determine if:
390 ·
π rad 180
=
13π 6
Solution: unit of measurement of angles; equivalent to the measurement of the angles created by 1/360 of a circle.
π = 3.14 A. True B. False ◦
30. Convert 10
26. Convert 2 22. Convert 1 rad
2·
Solution: 1·
pg. 2
◦
180 πrad
Solution:
Solution: ◦
180 πrad
◦
≈ 114.59
10 ·
π rad 180
=
π rad 18
≈ .175rad
◦
≈ 57.2956
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2007-2011 MathHands.com v.1010
