CHAPTER 1

Angles and their measurement Any study of trigonometry requires a basic understanding of the geometry and measurement of angles. In this chapter, I build this foundation. In Section 1.1 I introduce terminology for identifying the geometrical components of angles, and explain degree measurement for quantifying the their sizes. Radian measure, presented in Section 1.2, is a practical and convenient alternative to degree measure used many trigonometric settings. I conclude this chapter (Section 1.3) with a more general discussion of radian measure. I also present a formula for converting from one measure to the other. In the final section of this chapter I present some applications of angle measurement and arc-length.

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1.1 The concept of an angle

B

1.1.1

A

Figure 1.1: The line segment AB. B e id

rm Te

s al in

O

Init

ial s ide

A

Figure 1.2: Example of a positively oriented angle.

Definition of an angle

The shortest path between two points in a plane determines a line segment. The path connecting the points A and B in Figure 1.1 is the line segment determined by these points. The expressions AB and | AB| are notations for this segment and its length respectively. When appropriate, a dimension such as centimeters accompanies measurement of length. Two line segments that join at a common endpoint, called a vertex, form an angle. The line segments OA and OB in Figure 1.2 form an angle with vertex O. The two line segments, or sides, of an angle might have different lengths. In many trigonometric settings it is convenient to describe an angle by rotating a line segment about one of its endpoints so that it comes to rest on a second line segment. The vertex of the angle is the endpoint used for the center of the rotation. The curved, blue arrow in Figure 1.2 indicates that rotating OA about O in a counter-clockwise direction to OB describes the given angle. Angles described using a counter-clockwise rotation are positively oriented. The initial side of the angle is the initial position of the rotated line segment. The ter-

essentials of trigonometry

minal side is the terminal position of the rotated line segment. The initial and terminal sides of the angle in Figure 1.2 are OA and OB respectively. An angle is negatively oriented if a line segment is rotated in a clockwise direction about one of its endpoints. The curved arrow in Figure 1.3 indicates that the angle is negatively oriented. The base of the arrow lies on the initial side OA of the angle and the head lies on its terminal side OB. Normally, I will avoid labeling the initial and terminal sides of angles, and rely on curved arrows to indicate these sides. Example 1.1. The bases and heads of both blue arrows depicted in Figure 1.4 touch OB and OA respectively. Consequently, the initial side of the two angles is OB and the terminal side both is OA. You should notice that the angle with the smaller arrow is negatively oriented and the other is positively oriented. Two angles have the same geometric appearance if they have the same sides regardless of orientation. The four angles in figures 1.2 to 1.4 have the same geometric appearance even though they have different orientations. Angles have the same the initial and terminal sides are Co-terminal angles. The two angles in Figure 1.4 are co-terminal, but they are not co-terminal with the angles in Figure 1.2 or Figure 1.3. Co-terminal angles always have the same geometric appearance, but

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B e id

ls na

O

i rm e TIn iti

al s ide A

Figure 1.3: Example of a negatively oriented angle. B

O A Figure 1.4: Angles with the same geometric appearance as those in Figures 1.2 and 1.3.

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angles can have the same geometric appearance without being coterminal. Example 1.2. The five angles in Figure 1.5 have the same geometric appearance. The angles (a) and (c) are co-terminal, as are the angles (b) and (d). There is insufficient information to determine the initial and terminal sides of (e) so it is impossible to make any decision about its co-terminality with other angles. Figure 1.5: Angles for Example 1.2.

β α

O γ

Figure 1.6: Angles with different magnitudes.

1.1.2

O

O

O

O

O

(a)

(b)

(c)

(d)

(e)

Degree measure

The discussion in the previous paragraph suggests that any measure of an angle should specify its orientation. The angles α and β in Figure 1.6 have the same (positive) orientation, but the lengths of the curved arrows indicate that the rotations required to construct them have different magnitudes. The magnitude of an angle is independent of its orientation. The magnitude of the negatively oriented angle γ is

essentials of trigonometry

larger than the magnitude of α but less than the magnitude of β. This means that a second, necessary component of an angle measure is a unit to quantify its magnitude. The degree1 is a familiar unit used for this purpose. I will use the term measure to quantify both the magnitude and orientation of an angle. In this section, angle measures use degrees to describe the magnitude. If the measure is a positive number, the angle is positively oriented. If the measure is a negative number, the angle is negatively oriented.

The symbol ◦ is used to denote degree measurement. For example, 295◦ is read 295 degrees. A second notation for this unit is deg, so 295 deg = 295◦ . 1

1◦

O

A one degree (1 ◦ ) defines a positively oriented angle with magnitude of rotation of 1/360th of one complete revolution of a line segment about one of its endpoints. Figure 1.7 displays a positively oriented, one-degree angle. Example 1.3. You can envision an angle of measure −1◦ by mentally reversing the direction of the curved arrow in Figure 1.7.

5

Figure 1.7: A positively oriented 1◦ angle.

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P

O ◦ 360

A

Figure 1.8: Constructing a circle using a 360◦ angle. 0◦

O

Figure 1.9: A zero-degree angle. 180◦ B

O

A

Figure 1.10: Straight angle.

Since a 1◦ angle is 1/360th of a complete revolution of a line segment about one of its endpoints, one complete revolution of a line segment about one of its endpoints forms exactly 360 non-overlapping, one-degree angles with a common vertex. That is, one complete revolution of a line segment in the positive direction forms an angle with measure 360◦ or a full angle. The positively oriented angle in Figure 1.8 has measure 360◦ and represents one complete rotation of OA about O in the positive direction. The initial and terminal sides of a full angle coincide. Tracing the point A as you rotate OA about O in this construction creates a circle with radius |OA|, indicated by the (dashed) circle in Figure 1.8. This is the motivation behind the statement “there are 360 degrees in a circle.” Example 1.4. You can envision a −360◦ angle by mentally reversing the direction of the curved arrow in Figure 1.8. Tracing the point A in this case would draw the circle in the opposite direction. You should not confuse a full angle with the zero-degree angle shown in Figure 1.9. The zero-degree angle has no orientation. Consequently, it has no initial and terminal sides. There are other important, positively oriented, easily constructed angles. For example, 1/2 of a complete revolution of a line segment about one of its endpoints forms a straight or 180◦ angle. As shown in Figure 1.10, a straight angle lies along a line segment. A rotation

essentials of trigonometry

of 1/4th of a complete revolution forms a right or 90â—Ś angle. (See Figure 1.11.) A right angle bisects a straight angle. Graphically, you can indicate right perpendicular angles by half of a square as shown in Figure 1.12. You can use a protractor such as the one shown in Figure 1.13 to construct new angles and to measure given angles. The next example illustrates the use of a protractor.

B

90â—Ś O

A

Figure 1.11: Right angle. 100 110 120 60 50

140 40

80 70

130

80 70

90

60

100 110

50

120 130

Figure 1.12: Examples of right angles.

40 140

150

30 30

150

160

20 20

170

10

180 0

160

170

10

180 0

Figure 1.13: A protractor.

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Example 1.5. This example demonstrates the construction of a 60â—Ś angle angle using a protractor. Draw a line segment for one side of the angle and choose an endpoint of this line for the vertex O of the angle as shown in Figure 1.14(a). Place the arrow ( ) located on the base of the protractor2 on the vertex, ensuring that the side of the line lies along the edge as shown in Figure 1.14(b). Draw a small dot on your paper located just beyond the tick mark labeled 60 on the outer scale of the protractor as shown in the magnified region of Figure 1.14(b). Move the protractor away from your drawing and then construct the angleâ€™s second side by drawing the line segment that has the vertex as one endpoint and passes through the dot you just drew (See Figure 1.14(c).) to produce the angle shown in Figure 1.14(d).

40 150 30

(a)

30

0

10

20

0

0 18

17

16

0

0

15

0

14

0

13

0

70

12

0

80

11

0

10

90

0

10

80

0

70

50

20

O

10

O

0

0

(b)

18

130

0

17

0

11 0

0

12

180 0

120

14

10

30

110

O

13

20

170

50

60

30

160

0

100

60

40

150

10

180 0

50

140

15

O60 50

140

P

70

130

170

20

70 130

30

160

60 120

50

150

80

70

110

40

120

80

100

0

140

80

16

90

8090

70 60

40

110 120 130

P

100

100 110

60

50

40

Figure 1.14: Using a protractor to construct a 60â—Ś angle.

P

Some protractors may use a different indicator such as a small hole that you place over the vertex to designate this location. 2

P

8

40

(c)

140

30

150

0

20 20

160

(d)

essentials of trigonometry

Actually, the measure of the angle in Figure 1.13 is ±60◦ depend100 80 70 90 ing on the orientation chosen. When you construct an angle using a 120 110 60 80 100 70 110 130 50 protractor, you are responsible for indicating the orientation. 60 120 50 130 140 40 You can measure a given angle in a graphic by placing a protractor 40 140 150 30 correctly on the graphic—one side along the base of the protractor30 150 20 with the arrow on the vertex, and then locating the tic marks 160 thorough 20 160 −95◦ 170 10 170 10 which the second side passes. You can use this technique to ensure ◦ O 95 180 0 180 0 that angles you construct using a protractor are correct. Example 1.6. Figure 1.15 demonstrates the use of a protractor to measure a ±95◦ angle. Notice that the initial side of the −95◦ angle does not lie along the base of the protractor.

1.1.3 Angles with large magnitude Angles can have magnitudes larger than 360◦ . For example, I constructed the 400◦ in Figure 1.16 by revolving the initial side one complete revolution (360◦ ) and then an additional 40◦ . This produces the same result as forming a 40◦ angle and then revolving for a full angle. The 400◦ and 40◦ angles are co-terminal. More generally, adding positive integer multiples of 360◦ to positively oriented angles form new, positively oriented angles co-terminal with the original. A similar statement holds for adding positive integer

Figure 1.15: Using a protractor to measure a ±95◦ angle.

O

40◦

400◦

Figure 1.16: A 400◦ angle.

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multiples of −360◦ to negatively oriented angles. In this example, Figure 1.17, and Table 1.1 Red designates integer multipliers, gray designates the measure of the original angle, and blue designates the constructed angles. 3

If the initial and terminal sides agree and the angle has non-zero measure as in Figure 1.8, then the two endpoints of the spiral arrow count as a single cross. 4

Example 1.7. I constructed3 the angles represented in Figure 1.17 by adding k × (+360◦ ), k = 1, 3, to +85◦ and k × (−360◦ ), k = 2, 4, to −85◦ . (See the sub-figure captions.) The figures show that the four angles have the same geometric appearance. Angles (a) and (c) are coterminal, as are (b) and (d). You can determine the number of complete revolutions each larger angle makes by counting the number of times its associated spiral arrow crosses4 (not touches) its initial (or terminal) side. You should also notice the number of complete revolutions of an angle is the same as the integer multiple k used to construct it.

−85◦

85◦

−805◦

445◦

(a) 445◦ = +85◦ + 1 × (+360◦ )

−85◦

85◦ 1165◦

−1525◦

(b) −805◦ = −85◦ + 2 × (−360◦ ) (c) 1165◦ = +85◦ + 3 × (+360◦ ) (d) −1525◦ = −85◦ + 4 × (−360◦ )

Figure 1.17: Angles constructed using ±85◦ angles.

essentials of trigonometry

The expressions in the captions of the sub-figures in Figure 1.17 are reminiscent of the Remainder Theorem. As shown in the margin, dividing +445◦ by +360◦ gives a quotient of 1 and a remainder of +85◦ . Likewise, dividing −805◦ by −360◦ gives a quotient of 2 and a remainder of −85◦ . Table 1.1 summarizes these observations for the angles in Figure 1.17. Using the remainder theorem in this way facilitates the construction of angles with large magnitudes because the general appearance of the angle determined by the remainder is the same as the larger angle. The next example further illustrates this application of the Remainder Theorem. Original angle

Division by

Quotient

Remainder

+ 445◦ − 805◦ +1165◦ −1525◦

+360◦ −360◦ +360◦ −360◦

1 2 3 4

+85◦ −85◦ +85◦ −85◦

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Calculation 1 360◦ 445◦ 360◦ 85◦

2 −360◦ −805◦ −720◦ −85◦

Table 1.1: Determining the number of complete revolutions of angles.

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Example 1.8. Dividing 1125 ◦ by +360 ◦ yields a quotient of 3 with a remainder of 45 ◦ so that 1125 ◦ = 3(+360 ◦ ) + 45 ◦ 45◦ 1125◦

Figure 1.18: Determining the geometric appearance of an angle.

Hence, the angle with measure 1125 ◦ is co-terminal to the angle with measure 45 ◦ . The larger angle crosses its initial and terminal sides three times. Both angles are positively oriented. (See Figure 1.18.) You have seen that it is not difficult to derive an angle that completes a specified number of complete revolutions and is co-terminal with a given angle. The next example demonstrates a technique for determining an angle, if one exists, within a specified range with the same geometric appearance as a given angle.

essentials of trigonometry

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Example 1.9. In this example I construct an angle β within the range 1800◦ < β < 1900◦ ,

(1.1)

that is co-terminal with the angle of measure 85◦ . I need to determine the number k of complete revolutions the initial side of β makes before aligning along the terminal side of an angle of the 85◦ angle. To do this replace β in equation Equation 1.1 with k(360◦ ) + 85◦ and solve the resulting inequality for k as follows 1800◦ ◦

1800 − 85

◦ ◦

1715 1715◦ 360◦ 4.764

< k(360◦ ) + 85◦ < 1900◦ ◦

◦

< k(360 ) < 1900 − 85

◦

< k(360◦ ) < 1815◦ 1815◦ < k< 360◦ < k < 5.042

Consequently, k = 5 since k must be an integer, so β = 5(360◦ ) + 85◦ = 1885◦ . The angle β makes five complete revolutions in the positive direction. Figure 1.19 is a sketch of this angle.

+85◦ 1885◦

Figure 1.19: The angle with measure 1885◦ .

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+85◦ −275◦ Figure 1.20: Constructing an angle with opposite orientation.

Adding a positive integer multiple of −360◦ to a positively oriented angle angle may produce a negatively oriented angle co-terminal with the original. Examine, for example, the angles shown in Figure 1.20 and observe that −275◦ = 1 × (−360◦ + 85◦ ). Again, a similar statement holds when you add a positive integer multiple of 360◦ to a negatively oriented angle. These observations verify that two angles α and β are co-terminal if and only if β − α = k(360 ◦ ) for some integer k. Notice that the previous equation is equivalent to β = α + k(360 ◦ ). Example 1.10. The angles 210 ◦ and −510 ◦ are co-terminal since 210 ◦ − (−510 ◦ ) = 720 ◦ = 2(360 ◦ ). Notice that −510 ◦ = 210 ◦ − 2(360 ◦ ). You can use the method in Example 1.9 in more general settings as illustrated in the next example.

essentials of trigonometry

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Example 1.11. I will determine a negatively oriented angle β constrained by −1600◦ > β > −1800◦ with the same geometric appearance as an angle with measure 85◦ . To determine k, set β = k(−360◦ ) + 85◦

(1.2)

and solve the inequality

−1600◦ > k(−360◦ ) + 85◦ > −1800◦ to obtain (I reversed the inequalities because I divided by −360◦ .)

−1685◦ −1885◦ < k < . −360◦ −360◦

This last inequality reduces to

4.681 < k < 5.236. Hence, k = 5 since k must be an integer. Substituting this value into Equation 1.2 gives β = −1715◦ . Figure 1.21 shows that the initial side of the angle with measure −1715◦ completes four (5 − 1) revolutions before coming to rest on its terminal side. Why does the calculation

−1715◦ = 4(−360◦ ) − 275◦ verify this observation?

+85◦ −1715◦

Figure 1.21: The angle with measure −1715◦ .

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1.1.4 90◦ β α (a) α and β are complementary angles.

180◦ γ

α

(b) α and β are supplementary angles.

Figure 1.22: Complementary and supplementary angles.

Complementary, supplementary, acute, obtuse, and reflex angles

Two positively oriented angles and are complementary if the sum of their measures is 90 ◦ . They are supplementary if their measures sum to 180 ◦ . Figure 1.22 provides a geometric interpretation of these terms. Example 1.12. Angles with measures 55◦ and 35◦ are complementary since 55◦ + 35◦ = 90◦ . In fact, the angles α and β in Figure 1.22(a) have measures 55◦ and 35◦ respectively. Similarly, 55◦ and 125◦ are supplementary because 55◦ + 125◦ = 180◦ . The angle γ in Figure 1.22(b) has measure 125◦ . Example 1.13. Since 90◦ − 89◦ = 1◦ , the one-degree angle in Figure 1.7 complements the 89◦ angle. Similarly, the one-degree angle supplements the 179◦ angle. Other useful terms, described in Table 1.2, include acute, obtuse, and reflex angles. Term Degree measurement

Table 1.2: Measurement and examples of acute, obtuse, and reflex angles.

Example

Acute

Obtuse

Reflex

0◦ < ∠ < 90◦

90◦ < ∠ < 180◦

180◦ < ∠ < 360◦

essentials of trigonometry

Example 1.14. A 55 ◦ angle is acute and a 125 ◦ angle is obtuse. Any two complementary angles are acute. One angle of a supplementary pair is acute and the other is obtuse unless both angles in the pair have measure 90 ◦ . A 210 ◦ is a reflex angle. The inequalities in Table 1.2 are strict. This means that a right angle is neither an acute nor an obtuse angle, a straight angle is neither an obtuse nor a reflex angle, and a full angle is not a reflex angle5 . Example 1.15. To find an obtuse angle β that has the same general appearance as an angle with measure −945 ◦ , set −945 ◦ = β + k (360 ◦ ) or, equivalently, β = −945 ◦ − k(360 ◦ ). Since β is obtuse, the expression on the right of this last equality must satisfy 90 ◦ < −945 ◦ − k(360 ◦ ) < 180 ◦ ). Solve this last inequality to obtain 2.63 < k < 3.13. Choose k = 3 so that β = −945 ◦ + 3(360 ◦ ) = 135 ◦ .

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Some sources include a full angle in the definition of reflex angles 5

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Problem set for Section 1.1 1. For each angle below, label the initial and terminal sides. (a) O

B

(b) B

O

A

A

2. Use a protractor to estimate the measure of each of the following angles. Problem Set 1.1

(a)

1. (a) Initial: OB Terminal: OA (b) Initial: OA Terminal: OB 2. (a) - (d) 30◦ ,135◦ ,−67◦ ,210◦

(b)

O O (c)

O

(d) O

essentials of trigonometry

3. Use a protractor to construct angles with measure (a) 80◦

(b) −100◦

(c) 500◦

4. Determine if each pair of angles is co-terminal. (a) 72 ◦ and 792 ◦

(b) 483 ◦ and 123 ◦

(b) 24◦

(c) 125◦

(b)

(c)

80◦

O (b)

O

−100◦

(d) −37◦

6. Label each of the following angles as acute, obtuse, reflex, right, straight, full, or none of these. (a)

3. (a)

(c) 76 ◦ and 636 ◦

5. Find the complementary and supplementary angle measures for each of the following. Answer "None" if no such angle exists. (a) 73◦

Problem Set 1.1 (cont.)

(d)

(c) O 500◦ 4. (a) Yes

(e)

(f)

(g)

(h)

(b) Yes (c) No 5. (a) C: 17◦ S: 107◦

7. What angle is its own supplement?

(b) C: 66◦ S: 156◦ (c) C: None S: 55◦ (d) C: None S: None

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8. Using the strategies presented for the construction of Table Table 1.1, determine the missing values in the following table. Original Division Remainder Quotient angle by

6. (a) obtuse (b) obtuse (c) reflex (d) acute

+760◦ +566◦

(e) none (f) right (g) acute

−1248◦ −1994◦ −456◦

(h) straight 7. 45◦ 8. OA

+760◦ +566◦ +1588◦ −1248◦ −1994◦ −456◦

÷

+360◦ +360◦ +360◦ −360◦ −360◦ −360◦

Q

R

2 1 4 3 5 1

+40◦ +206◦ +148◦ −168◦ −194◦ −96◦

9. (a) 5, positive, α = 16◦ , β = −344◦

+360◦ +360◦ +360◦ −360◦ −360◦

4 3

+148◦

1

−96◦

9. How many complete revolutions does the initial side of angles with measurement given below make before coming to rest at its terminal side? Indicate the orientation of the rotation. Determine angles α, 0◦ ≤ α < 360◦ , and β, 0◦ ≥ β > 360◦ , that have the same geometric appearance as the original angle. (a) 1588◦

(b) −645◦

(c) 2520◦

10. Recall that two angles have the same geometric appearance if they have the same sides—the initial and terminal sides may be reversed. For each problem below, determine the angle β that satisfies the given conditions and has the same geometric appearance as the

essentials of trigonometry

given angle α. (a) α =

25◦ ,

1100◦

<β<

1200◦

(b) α = 45◦ , −360◦ < β < 0◦

(c) α =

−900◦ ,

0◦

<β<

360◦

(d) α = −175◦ , 180◦ < β < 270◦

11. You may encounter angles that contain decimal parts such as 4.5◦ . A one degree angle can be divided into 600 . The unit (’) represents a minute. Consequently, 4.5◦ = 4◦ 300 . You can convert any fractional part of a degree to minutes simply by multiplying the fractional part by 600 . In a similar way, a minute can be divided into 6000 or 60 seconds. Represent each angle below in terms of degrees, and minutes, and seconds. (a) 70.1◦

(b) 82.15◦

(c) 25.05◦

(d) −14.17◦

12. You can convert any quanity in seconds to a (possibly) decimal expression in minutes by dividing the given quanity by 60. You can use a similar technique to convert minutes to degrees. You should convert seconds to minutes before converting minutes to degrees. Represent each angle below in terms of degrees only. Display three decimal places in your answers. (a) 32◦ 500

(b) −127◦ 240

(c) 0◦ 140 3000

(d) 12◦ 530 + 6◦ 180

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(b) 1, negative, α = 75◦ , β = −285◦

(c) 7, positive, α = β = 0 (b) - (d) −45◦ ,805◦ ,180◦ ,185◦

11. (a) - (d) 70◦ 60 , 82◦ 90 , 20◦ 30 , −14◦ 100 1200 12. (a) - (d) 32.833◦ , −127.4◦ , 0.241◦ , 19.187◦

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1.2 Radian measure 1.2.1

1 rad O Figure 1.23: A 1 rad angle.

Basic properties and definition of the radian

In the previous section, you learned to use the degree unit to measure and construct angles. This section introduces the radian unit for quantifying angles. The notation “ rad” denotes the radian unit, so you interpret 1 rad as one radian and 2.7 rad as 2.7 radians. The angle in Figure 1.23 has measure 1 rad. You can use this figure to construct a mental images of angles such as 2 rad and 0.5 rad. Familiarity with the radian unit is essential to understanding and applying trigonometry. You should assume that the units are radians if none accompanies the measure of an angle. Consequently, the two statements “an angle has measure t rad” and “an angle has measure t” mean the same the thing. Omitting the radian unit when working with radian measure is more common than including it; normally, I include the unit for clarification purposes only. I will use the notation ∠AOB to describe the angle determined by the line segments OA and OB with initial side OA. This notation can be ambiguous since it does not specify the number of complete revolutions of the initial side before coming to rest on the terminal side of the angle. The context will always make this clear.

essentials of trigonometry

The lengths of arcs on a circle determine the radian measures of angles. In this section, I restrict this definition to the unit circle, or circle of radius one unit. I extend the definition of radian measure to circles of radius greater than one in Section 1.3. The unit (inches, meters) used to quantify the length of the radius is important in most applications but but may be omitted in general discussions. _ The red arc labeled AB on the unit circle in Figure 1.24 determines or subtends the ∠AOB. Conversely, the ∠AOB determines or subtends _ _ AB. The ∠AOB has radian measure t if AB has length t. That is, ∠AOB _ has measure t if the arc length of AB, the distance as measured along the red path, is t units. The angle in Figure 1.23 would subtend an arc on the unit circle of length 1 unit. Recall that the circumference of the unit circle is 2π. This means that the arc length of the arc subtended on the unit circle by a full angle with with vertex at the origin is 2π. The arc length of a straight angle has measure π rad because it subtends an arc that is half the arc length of the unit circle. Likewise, a right angle has measure π/2 rad. Example 2.1. An angle of measure π/4 rad subtends an arc on the unit circle of length π/4 or 1/2th of the length of the arc on the unit circle subtended by a right angle. Figure 1.25 is a geometric representation of this angle. You can construct an angle of measure π/8 by bisecting the angle in Figure 1.25.

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y B

_ AB

t O

1

A

x

Figure 1.24: The unit circle with an angle of measure t rad. B

_

π 4

O

AB A

Figure 1.25: Angle with measure π4 rad.

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y

O

1

x

Figure 1.26: Example of a non-central angle. y

O

1

Notice that I did not include the word â€œunitsâ€? or any other terms quantifying distances in the lengths of the arcs in Example 2.1. This is acceptable when no unit specified. However, if purposed problems include a unit such as feet, then this or a related unit (3 ft = 1 yd, for example) should accompany all lengths. For example, an angle with measure 6 rad subtends an arc with measure 6 ft or 2 yd on the unit circle. Likewise, if the length of the arcs on the unit circle is given in meters, then the radius of the unit circle is one meter, not one or one unit. The unit circle in Figure 1.24 is central because its center is at the origin O of the xy-plane. Similarly, the âˆ AOB in that figure is central because its vertex is at the origin O. This angle is also in standard position because it is central and its initial side lies along the x-axis. Centrality of an angles does not depend on their orientations. Figure 1.26 is an example of a non-central angle. Figure 1.27 is an example of a central angle not in standard position.

x

Figure 1.27: Example of a central angle that is not in standard position.

1.2.2

The radian protractor

In general, you would find it difficult to measure or construct angles using radian measure without a tool or software. A radian protractor would simplify this process. Evidently, radian protractors do exist but

essentials of trigonometry

I have only seen pictures of them. The graphic of a radian protractor in Figure Figure 1.28 provides some insight for the magnitude of various radian measurements. Notice, in particular, the size of the angle with measure 1 rad. You will learn a technique for using a degree protractor to construct or measure angles given in radians in Section 1.3.

1.8 2 2.2 2.4

1.6

1.4 1.2

π 2

1

π 3

2π 3

Figure 1.28: A radian protractor.

0.8

π 4

3π 4

0.6

2.6

π 6

5π 6

0.4

2.8

0.2 3

1 rad π

0

0

25

2.4

1.8 2 2.2

2π 3

26

thomas e. price

1.6

1.4

π 2

1.2

π 3

1 0.8

π 4

3π 4

0.6

π 6

π 6

0.4

0.2

π/6 rad 1 in

0

0

Figure 1.29: Using a radian protractor to construct an arc of length π/6 in.

Example 2.2. A central angle of radian measure π/6 rad subtends an arc of measure π/6 in on the circle of radius 1 in. As demonstrated in Figure 1.29, you can use a radian protractor to construct the angle, and you can use the inside, circular edge of the protractor to construct the arc subtended by this angle if you assume distance from the origin indicated by the arrow ( ) to the initial point of the base of the inner arc on the protractor is 1 in. If you assume that the distance from O to the outside edge of the protractor is 1 in, you can use the outside, circular edge to construct the arc.

1.2.3

Complementary, supplementary, acute, obtuse, and reflex angles in terms of radian measure

Two positively oriented angles are complementary if the sum of their measures is π/2 rad. The early treatment of trigonometry refereed arcs subtended on the unit circle by complementary angles as complementary arcs. Two positively oriented angles are supplementary if their sum is π rad. Example 2.3. The angles with measures π/6 rad and π/3 rad are complementary since π/6 + π/3 = π/2. The angle with measure π/4 rad is supplementary to 3π/4 rad. Unlike the degree protractor shown in Figure 1.13, the scales of

essentials of trigonometry

the outside and inside edges of the radian are not supplementary. Instead, the inside edge uses tick marks measured as proportions of π while the outside edge uses numerical values–no symbols such as π. A careful, visual examination of Figure 1.29 indicates that the depicted angle and arc have approximate measure .523 rad. Table 1.3 defines the terms acute, obtuse, and reflex angles using radian measure. Term Definition in radians

Acute 0<∠<

Obtuse π 2

π 2

<< π

Reflex π < ∠ < 2π

Example

1.2.4 Orientation of arcs A positively oriented angle subtends a positively oriented arc and a negatively oriented angle subtends a negatively oriented arc. Arcs always have the orientation of the angles that subtend them. Consequently, you may envision drawing positively oriented arcs in a counter-clockwise direction, and negatively oriented arcs in a clockwise direction. An arc is us usually drawn as a curved arrow lying

Table 1.3: Description of acute, obtuse, and reflex angles in radians with an example of each.

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thomas e. price

along a section of a circle to designate its orientation. O

A − π4

_

AB B

Figure 1.30: Angle with measure − π4 rad.

_

Example 2.4. The arc AB in Figure 1.30 is negatively oriented. You should note the role of the curved, red arrow lying along a section of the unit circle. Similar statements hold for the positively oriented arc _ labeled AB in Figure 1.25.

1.2.5

Arcs with large magnitude

Arcs may include one or more complete revolutions or wrappings of the circle. An arc with measure 2π units completes one revolution on the unit circle in the positive direction. An angle with measure −3π radian subtends a negatively oriented arc that wraps around the unit circle one and a half times. Coloring graphics of arcs with magnitudes larger than or equal to 2π provides no additional insight or visual information since entire the unit circle would be colored red in all these cases. In these cases I will not use color to help visually describe the arc. However, the blue, spiral arrows describing the angles serve to indicate the total number of complete revolutions of both angles and their associated arcs. Notice the similarity of the blue and red curved arrows in Figure 1.30. The next example illustrates the use of spiral arrows describing angles to provide insight to the wrappings of angles large magnitudes.

essentials of trigonometry

Example 2.5. The curved, blue arrow in Figure 1.31(a) indicates that the initial side of the angle with measure 11π/4 rad rotates around the circle one time, then rotates an additional 3π/4 rad (because 11π/4 = 2π + 3π/4) to its terminal side. This angle subtends an arc on the unit circle of measure 11π/4. The blue arrow describes the construction of this arc; it begins at the point (1, 0), makes one complete revolution in the positive direction on the unit circle, and then comes to rest at the point where the terminal side of the angle 11π/4 rad intersects the unit circle. Since −11π/4 = 1(−2π rad) − 3π/4, the initial side of an angle with measure −11π/ makes one complete revolution in the negative direction and then rotates an additional −3π/4 rad in the negative direction before landing on its terminal side. (See Figure 1.31(b).) As indicated by the blue arrow, the arc subtended by this angle begins at (1, 0) and completes one revolution around the unit circle in a clockwise direction before coming to rest at the point of intersection of the terminal side of the angle −11π/4 rad and the unit circle. Recall that for degree measure you could determine the number of complete revolutions of a positively oriented angle by dividing it by 360◦ . In the case of radian measure you should divide by 2π. This is not difficult to do symbolically. For example, to divide ]17π/3] rad by 2π, rewrite 2π = 6π/3 so that both the divisor and dividend have

29

y

O 11π/4 rad

x (1, 0)

(a)

y

−11π/4 rad O

x (1, 0)

(b)

Figure 1.31: Two arcs of length 11π/4 with different orientations.

30

thomas e. price

Calculation 2 6 π/3 17 π/3 12 π/3 5 π/3

17π 3

rad

the same denominator, 3. Then perform the division treating π/3 as a symbol as illustrated in the margin. I will use this calculation in the next example. Example 2.6. Since 17π/3 = 2(2π ) + 5π/3, the initial side of an angle with measure 17π/3 rad makes two complete revolutions in the positive direction and then rotates an additional 5π/3 rad in the positive direction before landing on its terminal side. Figure Figure 1.32 shows a geometric representation of this angle. The gray arc describes the angle with measure 5π/3 rad. The arc on the unit circle subtended by this angle would wrap around the circle twice and then continue along the circle until the point of intersection of the terminal side of the angle with the circle. If you have an angle with numerical measure, divide by a numerical approximation to 2π, say, for example, 6.28, a very rough approximation to 2π. Example 2.7. Because

Figure 1.32: A graph of the angle 5π/3 rad.

14.56 rad = 2(6.28 rad) + 2 rad, a 14.56 rad angle is co-terminal with the angle 2 rad and makes two complete revolutions before coming to rest on its terminal side. The arc subtended by this angle makes two complete revolutions around

essentials of trigonometry

the unit circle before coming to rest on its endpoint where the the unit circle intersects the terminal side of the angle. These numerical values in Example 2.7 are approximations because I used an 2π ≈ 6.28. The number 6.2832 is a better approximation. This yields 14.56 rad = 2(6.2832 rad) + 1.9936 rad, so the 14.56 rad angle is “more closely” co-terminal with the angle with measure 1.9936 rad. In most settings, arc length should never be a negative number. Both arcs portrayed in Figure 1.31 have length 11π/4. The arc in (a) is positively oriented and the arc in (b) is negatively oriented. Occasionally, it is beneficial to associate the orientation of an arc with its length. I will use the word measure to identify both orientation and arc length. The measure of the arcs in Figure 1.31 are 11π/4 and −11π/4 (units) respectively. Example 2.8. The measure of the arc on the central circle of radius 1 meter subtended by a central angle of −3 rad is −3 m. Of course, the arc has length 3 m.

31

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thomas e. price

Problem Set 1.2 1. (a) 3π/2 rad O

(b) −1 rad O

(c) 5π rad O

Problem set for Section 1.2 1. Construct angles with radian measures. Your images should display the orientations of the angles and the number of complete revolutions. (a) 3π/2 rad

(c) 5π rad

(e) 2 rad

(b) −1 rad

(d) π/4 rad

(f) π/3 rad

(a)

(b)

y

O

(f) π/3 rad

y

B

(d) π/4 rad

(e) 2 rad

(h) −π/3 rad

2. Determine the measure and length of the arc determined by the given angle in each figure.

5π rad

O

(g) −5π/4 rad

1.5 O

1 in A

x

B

3 O 1 cm A

x

essentials of trigonometry

(c)

(f)

y

Problem Set 1.2 (cont.)

y

B

B 3π 4

1 ft A

O

x

.6 x 1 A

O

O

(g) −5π/4 rad O

(d)

(g)

(h) −π/3 rad

y

y

O

B

B 13π 6

O 1 mm A

1 km

x

−4 O

A

x

(b) 3 cm, 3 cm (c) 3π/4 ft, 3π/4 ft (d) 13π/6 mm, 13π/6 mm (e) −1 yd, 1 yd

(h)

(e)

y

y

1 yd O

−1 A B

x

1 −5π 2

2. (a) 1.5 in, 1.5 in

O

(f) 0.6

x

(g) 4 yd (h)

−5π 2

33

34

thomas e. price

3. Construct arcs with given measure M on the circles with given radius R. What is the arc length L of your constructed arc.

3. (a) 1.5 ft

(b) M = −3π/4 mm, R = 1 mm

4. Construct a central angle of measure −5π/4 rad that has been rotated by π/4 rad. Is this angle central? Is it in standard position?

1.5 1 ft

5. Classify each of the following angles as acute, obtuse, reflex, or none of these.

(b) 1 mm

−3π/4

4. Yes, No.

(a) M = 1.5 ft, R = 1 ft

−3π/4 mm

(a) 1.2 rad

(c) 7π/8 rad

(e) π/2 rad

(b) −.5 rad

(d) 5π/2 rad

(f) 23π/12 rad

6. If possible, determine the measure of angles that satisfy the given conditions. (a) The angle makes one complete revolution before coming to rest on its terminal side, and has the same geometric appearance as the angle with measure. π/3 rad. (b) The reflex angle has the same geometric appearance as the angle with measure −3π/4 rad. .

(c) The negatively oriented angle makes three complete revolutions before coming to rest on its terminal side, and has the same geometric appearance as the angle with measure 5π/12 rad. .

(d) The acute angle has the same geometric appearance as the angle

essentials of trigonometry

with measure 25π/6 rad. .

y

(e) The angle has the same geometric appearance as the angle with measure 1 rad and makes one complete revolution. . (f) The negatively oriented angle has the same geometric appearance as the angle with measure π/2 rad and makes three complete revolutions. . (g) Two positively oriented angles areThe negatively oriented angle has the same geometric appearance as the angle with measure π/2 rad and makes three complete revolutions. . 7. If possible, determine the measure t, 13π/6 < t < 5π/2, of the angle that has the same geometric appearance as the angle with measure π/3 rad. 8. If possible, determine the measure t, −6π < t < 55π, of the angle that has the same geometric appearance as the angle with measure π/4 rad. 9. Arc length—not the measure of an arc—determines the distance traveled by a circle along a line. The following figure demonstrates that a rotation of −π rad (−2π rad) of a circle with radius 1 ft travels a distance of π ft (2π ft). The circle requires a rotation of −3 rad to travel a distance of 1 yd. How far does the circle travel if it completes 4.5 revolutions?

35

π/3

−5π/4 5. (a) Acute (b) None (c) Obtuse (d) None (e) None (f) Reflex 6. (a) 7π/3 rad (b) 5π/4 rad (c) 77π/12 rad (d) π/6 rad (e) 2π + 1 rad or approximately

x

36

thomas e. price

7.2832 rad (f) 2π + 1 rad or approximately 7.2832 rad (g) 2π + 1 rad or approximately 7.2832 rad 7. 7π/3 rad 8. −15π/4 rad 9. 9π ft

0 0 ft

−π π ft

−2π 2π ft

essentials of trigonometry

1.3

Radian measure on arbitrary circles, the conversion formula

1.4

Using angles and arc-length to solve problems

37

APPENDIX A

Solutions to Exercises and Chapter Quizzes

APPENDIX A. SOLUTIONS TO EXERCISES AND CHAPTER QUIZZES

40

210◦ = 180◦ + 30◦

Problem set for 1.1 (page 18) 1. (a)

(b)

l Initia

B

O Ter mi na

l Initia

O Ter m

A

l

B

10

160

ina

A

l

70

130

100 80

90

80 100

160

140 40 50

130

135◦

30 20

60

30

150 20

O

160

10

O

30◦

170

100 80

80

90

100

70

130

120 110

80◦

10

(b)

180 0

(c) - (b)

60 110

3. (a)

40 140

160

140

50 120

150

50

120

40

180 0

40 130

60

60 50

140

170

30

70 110

170

20

150 30

70

110

10

20

2. (a) - (b)

120

0 180

−67◦

170

150

O

0 180

O

−100◦

41

500◦ = 360◦ + 140◦

(c) O 500◦

4. (a) Yes, because 72 ◦ − 792 ◦ = −720 ◦ = 2(−360 ◦ ). (b) Yes, because

483 ◦ 76 ◦

− 123 ◦

− 636 ◦

=

360 ◦ .

560 ◦ ,

= (c) No, because not an integer multiple of −360 ◦ .

which is

5. (a) C: 17◦ = 90◦ − 73◦ S: 107◦ = 180◦ − 73◦ (b) C:

66◦

=

90◦

− 24◦

S:

157◦

=

180◦

− 24◦

(c) C: None, because 125◦ > 90◦ S: 55◦ = 180◦ − 125◦

have complements. S: None, because −37◦ < 0◦ Only positive angles with measure less than 180◦ have complements 6. (a) obtuse (b) obtuse (c) reflex (d) acute (e) none (f) none (g) acute (h) acute 7. 45◦ because 45◦ + 45◦ = 90◦ 8.

Row 1: Divide +760◦ by +360◦ to obtain a quotient of 2 and a remainder of +40◦ .

Row 2: Divide +566◦ by +360◦ to obtain a quotient of 1 and a remainder of 206◦ .

(d) C: None, because −37◦ < 0◦ Only positive angles with measure less than 90◦

Row 3: Divide +1588◦ by +360◦ to obtain a quotient of 4 and a remainder of +148◦ . Row 4: Divide −1248◦ by −360◦ to obtain a

42

APPENDIX A. SOLUTIONS TO EXERCISES AND CHAPTER QUIZZES

quotient of 3 and a remainder of −168◦ .

Row 5: Divide −1994◦ by −360◦ to obtain a quotient of 5 and a remainder of −194◦ .

Row 6: Divide −456◦ by −360◦ to obtain a quotient of 1 and a remainder of −96◦ .

9. (a) Five revolutions in the positive direction because 1816◦ = 5(360◦ ) + 16◦ . Hence, α = 16◦ and β = 16◦ − 360◦ = −344◦ (b) One revolution in the negative direction because −645◦ = 1(−360)◦ − 285◦ . Hence, β = −285◦ and α = 360◦ − 285◦ = 75◦ (c) Since 7(360◦ ) + 0◦ = 2520◦ , the angle forms seven complete rotations. The 0◦ = 2520◦ remainder verifies that α = β = 0. 10. (a) Set β = 25◦ + (k)360◦ and solve 1100◦ < 25◦ + (k)360◦ < 1200◦ to obtain 2.986 < k < 3.264. Hence, k = 3 so β = 25◦ + 3 × 360◦ = 1105◦ .

(b) Set β = 45◦ + (k )360◦ and solve −360◦ < 45◦ + (k)360◦ < 0◦ to obtain 1.99 < k < 2.26. Hence, k = 2 so β = 85◦ + (2)360◦ = 805◦ . (c) Set β = −900◦ + (k)360◦ and solve 0◦ < 805◦ + (k )360◦ < 360◦ to obtain 2.5 < k < 3.5. Hence, k = 3 so β = −900◦ + (3)360◦ = 180◦ . (d) Set β = −175◦ + (k)360◦ and solve 180◦ < −175◦ + (k)360◦ < 270◦ to obtain 0.99 < k < 1.24. Hence, k = 1 so β = −175◦ + (1)360◦ = 185◦ . ◦ ◦ 0 11. (a) Since .1(60) = 6, 70.1 = 70 6 . (b) Since .15(60) = 9, 82.15◦ = 82◦ 90 . (c) Since .05(60) = 3, 20.05◦ = 20◦ 30 . (d) Since .17(60) = 10.2, −14.17◦ = −14◦ 10.20 Further, since .2(60) = 12, −14.17◦ = −14◦ 100 1200 . ◦ 0 ◦ 12. (a) Since 50 ÷ 60 = 8.333 . . ., 32 50 = 32.833 (. ◦ 0 ◦ (b) Since 24 ÷ 60 = .4, −127 24 = −127.4 .

43

0◦ 140 3000

14.50 .

(c) Since 30 ÷ 60 = .5, = Next, 14.5 ÷ 60 = .241 . . ., so 0◦ 140 3000 = 0.241◦ . ◦ ◦ 0 ◦ 0 (d) Since 12 + 6 18 = 18 71 and 71 ÷ 60 = 1.183 . . ., 12◦ 530 + 6◦ 180 = 19.183◦

(d) O

Problem set for 1.2 (page 32) (e) O

O 1. (a) O

(f) O

(b)

O

(g)

O (c)

O

5π rad

(h)

44

APPENDIX A. SOLUTIONS TO EXERCISES AND CHAPTER QUIZZES

2. (a) The measure and length of this arc is 1.5 in. Remember that an angle radian measure always subtends an arc with the same unit as the radius. (b) The measure and length of this arc is 3 cm. Remember that an angle radian measure always subtends an arc with the same unit as the radius. (c) The measure and length of this arc is 3π/4 ft. Remember that an angle radian measure always subtends an arc with the same unit as the radius. (d) The measure and length of this arc is 13π/6 mm ft. Remember that an angle radian measure always subtends an arc with the same unit as the radius. (e) Since the arc is negatively oriented it has measure −1 yd. The arc has length 1 yd.

(f) The arc length is .6. You may assume that the dimension is radian since none is given for the radius of the circle. (g) Since the arc is negatively oriented it has measure −4 yd so its arc length is 4 yd. (h) Since the arc is negatively oriented it has 5] pi measure −25π , so its arc length is 2 .The arc −5] pi

length is 2 . You may assume that the dimension is radian since none is given for the radius of the circle. 3. (a) Since the radius of the circle is R = 1 ft, a positively oriented angle of 1.5 = 1.5 rad subtends an arc with measure M = 1.5 ft. The arc length of this arc is L = 1.5 ft.

45

y

y 1.5 ft 1.5

O

1 ft

O

x

1 mm

x

−3π/4 −3π/4 mm

(b) Since the radius of the circle is R = 1 mm, an angle with measure −3π/4 subtends an arc with measure M = −3π/4 mm. The arc has length L = 3π/4 mm.

4. The constructed angle is central but not in stany

π/3

x

−5π/4 dard position.

x

5. (a) Acute, since 0 rad < 1.2 rad < π/2 rad.

46

APPENDIX A. SOLUTIONS TO EXERCISES AND CHAPTER QUIZZES

Observe that 0 < 1.2 < 1.57 u π/2. (b) None, since the angle has negative measure. Acute, obtuse, and reflex angles have positive measure. (c) Obtuse, since π/2 rad < 7π/8 rad < π rad. Observe that pi/2 < π/2 + 3π/8 = 7π/8 < π/2 + π/2 = π. (d) None, since 5π/2 rad > 2π rad because 5π/2 = 4π/2 + π/2 > 2π. (e) None, since, according to Table 1.3, acute angles have measure less than π/2 and obtuse and reflex angles have measure greater than π/2. (f) Reflex, since, π rad < 23π/12 rad < 2π rad. Observe that π = 12π/12 < 23π/2 < 24π/12 = 2π.

6. (a) Add 2π to the given measure to obtain t = π/3 rad + 2π rad = 7π/3 rad. (b) Add 2π to the given measure to obtain t = −3π/4 rad + 2π rad = 5π/4 rad.

(c) Add 8π to the given measure to obtain t = 5π/12 rad + 2π rad = 77π/12 rad. (d) Since

1 25π ÷ 2π = 2 + 6 12 , the angle makes two revolutions. Multiply both sides of this last equation by 2π to obtain 25π π = 4π + 6 6 . Consequently, the given angle has the same geometric appearance an acute angle with measure π/6 rad. (e) Add 2π u 6.2832 to 1 to obtain the measure 2π + 1 rad.

47

(f) The desired angle has measure measure −8π + π/2 rad = −15π/2 rad.

(g) 7. Let k denote the number of complete revolutions the angle t makes. Since the desired angle has the same geometric appearance as the angle with measure π/3 rad, t = 2kπ + π/3. Hence, 13π 6 13 1 − 6 6 11 12

π 5π < 3 2 15π 1 2kπ < 6 6 13 k< 12

< 2kπ + < <

The only integer that satisfies the previous inequalities is k = 1, so, the measure of the angle t is 2π + π/3 rad = 7π/3 unit. 8. Let k denote the number of complete revolu-

tions the angle t makes. Since the desired angle has the same geometric appearance as the angle with measure π/4 rad, t = −2kπ + π/4. Hence,

−6π 23 4 23 8

< −2kπ + 19 4 19 k> 8

π < −5π 4

< 2k > <

The only integer that satisfies the previous inequalities is k = 2, so, the measure of the angle t is −4π + π/4 rad == 15π/4 unit. 9. The angle determined by 4.5 revolutions is 9π rad (4.5 × 2π). The length of an arc subtended by this angle on a circle of radius 1 ft is 9π ft. So the distance traveled by the circle is 9π ft.

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