Technology & Engineering
Radians vs Degrees
Radians and degrees are units of measurement for angles. Degrees are based on dividing a circle into 360 equal parts, while radians are based on the radius of a circle. Radians are often preferred in mathematical and engineering applications due to their simpler relationship with the circumference of a circle and trigonometric functions.
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7 Key excerpts on "Radians vs Degrees"
eBook - PDF- Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen(Authors)
- 2012(Publication Date)
- Wiley(Publisher)
The trigonometric functions began to be used on problems that had nothing whatsoever to do with angles and triangles. In this chapter we generalize the concept of trigonometric ratios along the lines just suggested. Before we undertake this task, however, we will introduce anoth- er form of angle measure called the radian. 2.1 DEGREES AND RADIANS • Degree and Radian Measure of Angles • Angles in Standard Position • Arc Length and Area of a Sector of a Circle ■ ■ Degree and Radian Measure of Angles In Chapter 1 we defined an angle and its degree measure. Recall that a central angle in a circle has angle measure 1° if it subtends an arc of the circumference of the circle. Another approach to measuring angles, radian measure, has advantages 1 360 52 2 TRIGONOMETRIC FUNCTIONS (c) 2 rad r r r (d) 3 rad r r r r 1 360 circumference 1˚ (a) (b) 1 rad r r FIGURE 1 Degree and radian measure 2.1 Degrees and Radians 53 The radian measure of a central angle of a circle is given by where s is the length of the arc opposite u and r is the radius of the circle. [Note: s and r must be in the same units.] u = s r radians u r s in mathematics and the sciences. Suppose that an arc of a circle has length r, equal to the circle’s radius. Then the central angle subtended by that arc has radian mea- sure 1 (see Fig. 1b). Similarly, if an arc of a circle has length 2r or 3r, then the central angle subtended by that arc has radian measure 2 or 3, respectively (see Fig. 1c and 1d). Note that an angle of radian measure 3 is smaller than a straight angle. The radian measure of any central angle u is defined by dividing the arc length s by the radius r. What is the radian measure of a central angle subtended by an arc of 32 cm in a circle of radius 8 cm? Remark The word radian or its abbreviation rad is sometimes omitted when we are deal- ing with the radian measure of angles. An angle of measure 4 is understood to mean 4 radians, not 4 degrees; if degrees are intended, we write 4°.
eBook - PDF- Ron Larson(Author)
- 2021(Publication Date)
- Cengage Learning EMEA(Publisher)
All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 4.1 Radian and Degree Measure 261 GO DIGITAL Radian Measure The amount of rotation from the initial side to the terminal side determines the measure of an angle. One way to measure angles is in radians. This type of measure is especially useful in calculus. To define a radian, use a central angle of a circle, which is an angle whose vertex is the center of the circle, as shown in Figure 4.5. The circumference of a circle is 2π r units, so it follows that a central angle of one full revolution (counterclockwise) corresponds to an arc length of s = 2π r. Moreover, 2π ≈ 6.28, so there are just over six radius lengths in a full circle, as shown in Figure 4.6. The units of measure for s and r are the same, so the ratio sr has no units—it is a real number. The measure of an angle of one full revolution is sr = 2π rr = 2π radians. Here are the measures of some other angles. 1 2 revolution = 2π 2 = π radians 1 4 revolution = 2π 4 = π 2 radians 1 6 revolution = 2π 6 = π 3 radians 1 8 revolution = 2π 8 = π 4 radians These and other common angles are shown below. π 6 π 4 π 3 π 2 π π 2 Recall that the four quadrants in a coordinate system are numbered I, II, III, and IV. The figure below shows which angles between 0 and 2π lie in each of the four quadrants. An angle whose terminal side lies on the x- or y-axis, such as 0, π 2, π , or 3π 2, is a quadrantal angle. The terminal sides of quadrantal angles do not lie within quadrants. Note that angles between 0 and π 2 are acute angles and angles between π 2 and π are obtuse angles.
No longer available |Learn more- Alfred Basta, Stephan DeLong, Nadine Basta, , Alfred Basta, Stephan DeLong, Nadine Basta(Authors)
- 2013(Publication Date)
- Cengage Learning EMEA(Publisher)
Construction of the Radian System of Angle Measurement In a circle of arbitrary radius r with center point O , draw a ray, OA , as shown in Figure 7.4. If the ray rotates on O to trace an arc, AB , which happens to have Figure 7.4 The definition of one radian © Cengage Learning 2014 A O r r B Arc AB H11005 r Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 260 Chapter 7 length exactly that of the radius, then the measure of angle AOB is defined as the unit angle in this system and is said to be one radian (rad). The measure of any angle in the radian system can be found by measuring the arc of the circle between the initial and terminal ray locations and dividing the length of that arc by the radius of the circle. Note that, whatever the units of measurement of arc and radius, creation of this ratio generates an angular mea-surement that is real number valued only and is otherwise free of units. Recalling that the circumference of a circle is given by C 5 2 p r , it follows immediately that the angle corresponding to a complete revolution of the circle measures 2 p radians. The same angle when expressed in degrees measures 360 ° , 360 ° and thus we have discovered a crucial equivalence that ties the two angle measurement systems together: These equalities allow us to expand our list of conversion factors and fur-ther our opportunity to convert between different units of measure: H9266 H9266 180 180 ° ° anduni00A0 are our new conversion factors EXAMPLE 7.3 Convert 2.456 radians to degrees.
eBook - PDF- Paul A. Calter, Michael A. Calter(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
Angle Conversion By definition, an arc having a length equal to the radius of the circle subtends a central angle of one radian. It follows that an arc having a length of twice the radius subtends a central angle of two radians, and so on. Thus an arc with a length of times the radius (the entire circumference) subtends a central angle of radians. Therefore radians is equal to 1 revolution, or This gives us the following conversions for angular measure: Angle Conversions 78 If we divide by , we get Some prefer to use Eq. 78 in this approximate form instead of the one given. You may also prefer to use Using Eq. 78, we convert angular units in the same way that we converted other units in Chapter 1. ◆◆◆ Example 1: Convert to radians and revolutions. Solution: By Eq. 78, Note that and in Eq. 78 are exact numbers, so we keep the same number of digits in our answer as in the given angle. Now converting to revolutions, we obtain ◆◆◆ ◆◆◆ Example 2: Convert 1.8473 rad to degrees and revolutions. Solution: By Eq. 78, 1.8473 rad a 360° 2p rad b 105.84° 47.6° a 1 rev 360° b 0.132 rev 360° 2p 47.6° a 2p rad 360° b 0.831 rad 47.6° p rad 180°. 1 rad 57.3° 2p 360° 1 rev 360° 2p rad 360°. 2p 2p 2p u u u u? u. 0°, 90 0 Degrees Radians (decimal) Radians (in terms of ) O 0 0 4 –– 2 –– 3 4 ––– 10 20 30 40 50 60 1 0.5 1.5 2.5 2 70 80 100 110 120 130 140 150 160 170 180 3 P a p e r strip Section 1 ◆ Radian Measure 407 and ◆◆◆ Angle Conversion by Calculator We showed how use a calculator to convert between decimal degrees and degrees, minutes, and seconds in Chapter 1. Here we will show conversions between deci- mal degrees and radians. ◆◆◆ Example 3: Convert 2.865 radians to degrees, on the TI-83/84 calculator. Solution: • Put the calculator into degree mode. • Enter the angle, in radians. On the TI-83/84, the radian symbol is found in the Angle menu.
eBook - PDF- Charles P. McKeague(Author)
- 2020(Publication Date)
- XYZ Textbooks(Publisher)
119 3 iStockphoto.com © ventdusud RADIAN MEASURE Chapter Outline 3.1 Reference Angle 3.2 Radians and Degrees 3.3 Definition III: Circular Functions 3.4 Length of an Arc and Area of a Sector 3.5 Velocities Summary Test In Section 3.2, we will show how to convert back and forth between degrees and radians. Once we are able to do these conversions, we can apply all the material we have developed for degrees to radians. There are some interesting relationships that exist between the trigonometric functions we have have defined previously and the points on the unit circle. We cover these relationships and some material on even and odd functions in Section 3.3. In the last two sections of this chapter, we will apply the material from the first part of the chapter to circles in general. We will begin by deriving formulas for arc length and area and then proceed on to define angular velocity for points in motion on the circumference of a circle. The most important material in the chapter, from the standpoint of what is needed to continue on through the book, is the material in the first three sections. These are the sections in which you will be introduced to, and become familiar with, radian measure. In Chapter 4, when we graph the different trigonometric functions, we will work almost exclusively in radian measure. I n Chapters 1 and 2, we used degree measure exclusively to give the measure of angles. We will begin this chapter with another kind of angle measure called radian measure. Radian measure gives us a way to measure angles with real numbers instead of degrees. As you will see, there are a number of situations that occur in trigonometry for which real numbers are the more appropriate measure for angles. 120 Success Skills The study skills for this chapter are about attitude. They are points of view that point toward success. 1.
eBook - PDF- Cynthia Y. Young(Author)
- 2021(Publication Date)
- Wiley(Publisher)
6.6 Radian Measure and Applications 581 In the above example the units, cm, canceled, therefore correctly giving radians as a unitless real number. Because radians are unitless, the word radians (or rad) is often omitted. If an angle measure is given simply as a real number, then radians are implied. Words Math The measure of θ is 4 degrees. θ = 4° The measure of θ is 4 radians. θ = 4 6.6.2 Converting Between Degrees and Radians 6.6.2 Skill Convert between degrees and radians. 6.6.2 Conceptual Understand that degrees and radians are both units for measuring angles. In order to convert between degrees and radians, we must first look for a relationship between them. We start by considering one full rotation around the circle. An angle corresponding to one full rotation is said to have measure 360°. Radians are defined as the ratio of the arc length that the angle intercepts on the circle to the radius of the circle. One full rotation corresponds to an arc length equal to the circumference of the circle. Words Math Write the angle measure (in degrees) that corresponds to one full rotation. θ = 360° Write the angle measure (in radians) that corresponds to one full rotation. Arc length is the circumference of the circle. s = 2πr Substitute s = 2πr into θ (in radians) = s _ r . θ = 2πr ____ r = 2π radians Equate the measures corresponding to one full rotation. 360° = 2π radians Divide by 2. 180° = π radians Divide by 180° or π. 1 = π ____ 180° or 1 = 180° ____ π This leads us to formulas that convert between degrees and radians. Let θ d represent an angle measure given in degrees and θ r represent the corresponding angle measure given in radians. Converting Degrees to Radians To convert degrees to radians, multiply the degree measure by π ____ 180° . θ r = θ d ( π ____ 180° ) Converting Radians to Degrees To convert radians to degrees, multiply the radian measure by 180° ____ π .
eBook - PDF- Richard Dodd(Author)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
4 Unit of angular measure (radian) 4.1 SI definition of the radian The radian is the angle subtended at the centre of a circle by an arc along the circumference whose length is equal to that of the radius of the circle. The dimension of angular measure is [L] . [L] −1 = [1], its unit is the radian and its symbol is rad. In Figure 4.1, the angle AOB is equal to one radian if −→ AB −→ OA = 1, where OA = OB is the radius of the circle and −→ AB is the distance along the circumference of the circle from A to B. Given that the circumference of a circle of radius r is 2πr , then the number of radians in the circle is simply 2πr/r or 2π . 4.2 Commonly used non-SI units of angular measure 4.2.1 Converting from ( ◦ ) and (h m s) to radians Positions of astronomical objects are regularly given as angles from a reference point or plane in degrees ( ◦ ), minutes ( ) and seconds ( ), or sometimes as a measure of time from a reference point in hours (h), minutes (m) and seconds (s). There are 60 in 1 and 60 s in 1 m, 60 in 1 ◦ and 60 m in 1 h and 360 ◦ or 24 h in a circle. This sexagesimal system was first used by the Babylonians (Pannekoek, 1961) more than 2000 years ago and most astronomical catalogues still use it, though some catalogues do include positions in radians, e.g., the SuperCOSMOS Sky Survey. 9 There are 2π rad in 360 ◦ , which gives the following conversion factors from sexagesimal measures to radians: 1 ◦ = 2π 360 = 0.017 453 292 520 rad 9 See www-wfau.roe.ac.uk/sss 30 4.2 Commonly used non-SI units of angular measure 31 O A B Figure 4.1.
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