Radians

10 Key excerpts on "Radians"

• 

  eBook - PDF

  Precalculus

  A Prelude to Calculus

  • Sheldon Axler(Author)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  A Natural Unit of Measurement for Angles We have been measuring angles in degrees, with 360 ◦ corresponding to a rotation through the entire circle. Hence 180 ◦ corresponds to a rotation through one-half the circle (thus generating a line), and 90 ◦ corresponds to a rotation through one-fourth the circle (thus generating a right angle). There is nothing natural about the choice of 360 as the number of degrees in a The use of 360 ◦ to denote a complete rotation around the circle probably arose from trying to make one day’s rotation of the earth around the sun (or the sun around the earth) correspond to 1 ◦ , as would be the case if the year had 360 days instead of 365 days. complete circle. Mathematicians have introduced another unit of measurement for angles, called Radians. Radians are often used rather than degrees because working with Radians can lead to much nicer formulas than working with degrees. The unit circle has circumference 2π. In other words, an ant walking around the unit circle once would walk a total distance of 2π. Because going around the circle once corresponds to traveling a distance of 2π, the following definition is a natural choice for a unit of measurement for angles. As we will see, this definition makes the length of an arc on the unit circle equal to the corresponding angle as measured in Radians. Radians Radians are a unit of measurement for angles such that 2π Radians correspond to a rotation through an entire circle. Radians and degrees are two different units for measuring angles, just as feet and meters are two different units for measuring lengths. Example 1 Convert each of the following angles to degrees. Then sketch the radius of the unit circle corresponding to each angle.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Analytic Trigonometry with Applications

  • 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°.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Trigonometry

  • Ron Larson(Author)
  • 2017(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  For example, in Exercise 68 on page 131, you will use angles and their measure to model the distance a cyclist travels. Connel/Shutterstock.com Copyright 2018 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. 1.1 Radian and Degree Measure 123 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 1.5. Definition of a Radian One radian (rad) is the measure of a central angle θ that intercepts an arc s equal in length to the radius r of the circle. (See Figure 1.5.) Algebraically, this means that θ = s r where θ is measured in Radians. (Note that θ = 1 when s = r.) 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 1.6. The units of measure for s and r are the same, so the ratio sH20862r has no units—it is a real number. The measure of an angle of one full revolution is sH20862r = 2π rH20862r = 2π Radians, so you can obtain the following. 1 2 revolution = 2π 2 = π Radians 1 4 revolution = 2π 4 = π 2 Radians 1 6 revolution = 2π 6 = π 3 Radians These and other common angles are shown below.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Precalculus with Limits

  • 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 sr has no units—it is a real number. The measure of an angle of one full revolution is sr = 2π rr = 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.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Using SI Units in Astronomy

  • 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.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Technical Mathematics with Calculus

  • Paul A. Calter, Michael A. Calter(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  The radian uses a part of the circle itself, (the radius) as a unit. This chapter also marks our return to trigonometry, with which we last studied right and oblique triangles. It will be followed by a few more chapters on trigonometry. 300°. 14 Pitch diameter = 22.5 mm Rack Pinion FIGURE 14–1 Rack and pinion. 406 Chapter 14 ◆ Radian Measure, Arc Length, and Rotation 90 0 10 20 30 40 50 60 70 80 100 110 120 130 140 150 160 170 180 P a p e r strip  O FIGURE 14–2 One radian r r r FIGURE 14–3 14–1 Radian Measure ■ Exploration: Try this. (a) Make a photocopy of a protractor, Fig. 14–2. (b) Cut a strip of paper whose length equals the radius of the protractor circle. (c) Starting at bend the strip around the edge of the protractor and mark its end point (this will be easier if you paste the photocopy to cardboard and cut it out). (d) Draw a line from the end of the strip to the center O, forming angle What is the significance of About how large is in degrees? Estimate roughly how many angles of size will fit completely around the circle. ■ The angle you constructed in the exploration is defined as one radian. A radian is the central angle subtended by an arc whose length is one radius, Fig. 14–3. We will see that the radian is a dimensionless ratio and not a unit of measure like the degree or the inch. Thus it is not strictly correct to write “Radians” after the angle, but we will do so anyway because it makes it easier to keep track of our an- gles during a computation. 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.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Trigonometry

  • Charles P. McKeague(Author)
  • 2020(Publication Date)
  • XYZ Textbooks

    (Publisher)

  As you will see, specifying the measure of an angle with radian measure gives us a way to associate the measure of an angle with real numbers, rather than degrees. A Radian Measure To understand the definition for radian measure, we have to recall from geometry that a central angle is an angle with its vertex at the center of a circle. FIGURE 1 θ O r r Angle θ has a measure of 1 radian The vertex of θ is at the center of the circle; the arc cut off by θ is equal in length to the radius Note It is common practice to omit the word radian when using radian measure. If no units are showing, an angle is understood to be measured in Radians; with degree measure, the degree symbol ° must be written. θ = 2 means the measure of θ is 2 Radians θ = 2° means the measure of θ is 2 degrees To find the radian of any central angle, we must find how many radii there are in the arc it cuts off. To do so, we divide the arc length by radius. If the radius is 2 centimeters and the arc cut off by central angle θ is 6 centimeters, then the radian measures of θ is _ 6 = 3 Radians. 2 Definition In a circle, a central angle that cuts off an arc equal in length to the radius of the circle has a measure of 1 radian. That is, in a circle of radius r , a central angle that measures 1 radian will cut off an arc of length r (Figure 1). 132 Chapter 3 Radian Measure Here is the formal definition: Definition If a central angle θ in a circle of radius r cuts off an arc of length s , then the measure of θ in Radians is given by _ s r . (See Figure 2.) FIGURE 2 θ O r s θ (in Radians) = s r 3.2 Videos Example 1 A central angle θ in a circle of radius 3 centimeters cuts off an arc of length 6 centimeters. What is the radian measure of θ ? FIGURE 3 θ 3 cm 6 cm Solution We have r = 3 centimeters and s = 6 centimeters, therefore, s θ (in Radians) = _ r 6 cen = ___________ timeters 3 centimeters = 2 We say the radian measure of θ is 2 or θ = 2 Radians.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Precalculus

  • Cynthia Y. Young(Author)
  • 2023(Publication Date)
  • Wiley

    (Publisher)

  This leads to a general definition of radian measure. s θ r r If a central angle θ in a circle with radius r intercepts an arc on the circle of length s, then the measure of θ, in Radians, is given by θ (in Radians) = s _ r Note: The formula is valid only if s (arc length) and r (radius) are expressed in the same units. Radian Measure Caution To correctly calculate Radians from the formula θ = s __ r , the radius and arc length must be expressed in the same units. 4.1 Angle Measure 355 In the previous example, the units, cm, canceled, therefore correctly giving Radians as a unitless 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 Note that both s and r are measured in units of length. When both are given in the same units, the units cancel, giving the number of Radians as a dimensionless (unitless) real number. One full rotation corresponds to an arc length equal to the circumference 2πr of the circle with radius r. We see then that one full rotation is equal to 2π Radians. θ full rotation = 2πr ___ r = 2π Concept Check What is the measure (in Radians) of a central angle θ that intercepts an arc of length A cm on a circle with radius A mm? Answer: 10 rad Caution Units for arc length and radius must be the same in order to use θ = s __ r Video EXAMPLE 2 Finding the Radian Measure of an Angle What is the measure (in Radians) of a central angle θ that intercepts an arc of length 6 centimeters on a circle with radius 2 meters? Common Mistake A common mistake is forgetting to first put the radius and arc length in the same units. Correct Write the formula relating radian measure to arc length and radius. θ (in Radians) = s _ r Substitute s = 6 centimeters and r = 2 meters into the radian expression.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Algebra and Trigonometry

  • 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   .

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Pre-Calculus All-in-One For Dummies

  Book + Chapter Quizzes Online

  • Mary Jane Sterling(Author)
  • 2023(Publication Date)
  • For Dummies

    (Publisher)

  As it turns out, the number 360 was picked to represent the degrees in a circle only for convenience. Chapter 8 IN THIS CHAPTER » Discovering alternate trig function definitions » Inserting triangles on the unit circle » Calculating trig functions on the unit circle 174 UNIT 3 The Essentials of Trigonometry What’s the convenience of the number 360, you ask? Well, you can divide a circle into many different, equal parts by using the number 360, because it’s divisible by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45 . . . and these are just the numbers less than 50! Basically, the number 360 is pretty darn flexible for performing calculations. The radian was introduced as an angle measure to make some of the computations easier and nicer. The word radian is based on the same root word as radius, which is the building block of a circle. An angle measurement of 360 , or a complete circle, is equal to 2 Radians, which breaks down in the same way that degrees do. In pre-calculus, you draw angles with their vertices at the origin of the coordinate plane 0 0 , , and you place one side on the positive x-axis (this side is called the initial side of the angle, and it is always in this location). The other side of the angle extends from the origin to anywhere on the coordinate plane (this side is the terminal side). An angle whose initial side lies on the positive x-axis is said to be in standard position. If you move from the initial side to the terminal side in a counterclockwise direction, the angle has a positive measure. If you move from the initial side to the terminal side in a clockwise direc- tion, you say that this angle has a negative measure. A positive/negative discussion of angles brings up another related and important point: co- terminal angles. Co-terminal angles are angles that have different measures, but their terminal sides lie in the same spot.

  Sign up to read

  Learn more about book