Radians vs Degrees

4 Key excerpts on "Radians vs Degrees"

• 

  eBook - ePub

  Mechanical Design for the Stage

  • Alan Hendrickson(Author)
  • 2012(Publication Date)
  • Routledge

    (Publisher)

  Figure 7.2 ). Regardless of the units used to measure the radius and arc length—feet, meters, or light years—the ratio cancels those units out and the result is a dimensionless number. So, for instance, angular speed will be described in terms of “radians per second” to unambiguously state how the angles are measured, but the true units of angular speed are simply “per second” or “1/sec”.

  Figure 7.2 Radian angle measurement

  One full revolution, or 360°, in radian measure is the ratio of the circumference of a circle of radius r to that radius, r. Since the circumference is 2πr, the ratio of arc length to radius for one revolution becomes (see right side of Figure 7.2 )

  This value allows two common conversions to be developed—degrees to radians

  d e g r e e s ×

  2 π   r a d i a n s   p e r   r e v o l u t i o n

  360   d e g r e e s   p e r   r e v o l u t i o n

  = d e g r e e s × 0.075 = r a d i a n s

  t u r n s × 2 π = t u r n s × 6.28 = r a d i a n s

  EXAMPLE: During a scene change, a turntable rotates 110° (the direction of rotation is intentionally being left out, for reasons to be explained below). What is its angular displacement in radians?

  SOLUTION: Since displacement is the subtraction of one position from another, or, stated differently, the difference between two positions, the exact value of the initial position at time t1

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  eBook - ePub

  Barron's Math 360: A Complete Study Guide to Pre-Calculus with Online Practice

  • Lawrence S. Leff, Christina Pawlowski-Polanish(Authors)
  • 2021(Publication Date)
  • Barrons Educational Services

    (Publisher)

  minute. The notation 28°30′ is read as “28 degrees, 30 minutes.” Since 60 minutes is equivalent to 1 degree, dividing 30 minutes by 60 changes 30 minutes to a fractional part of a degree. Thus:

  TO CHANGE MINUTES TO DEGREES

  MEASURING ANGLES IN RADIANS

  In Figure 9.1 , angle θ cuts off an arc of circle O that has the same length as the radius of the circle. Angle θ measures 1 radian. Unlike degrees, radians are real numbers.

  FIGURE 9.1 Defining a radian in a circle with radius r

  Since the total number of radii that can be marked off along the circumference of the circle is 2π, the radian measure of a circle is 2π. The degree measure of a circle is 360°. Hence, 2π radians = 360°, so π radians = 180°. This relationship provides the conversion factor for changing from one unit of angle measure to the other.

  RADIAN AND DEGREE CONVERSIONS An angle of 1 radian is the angle at the center of a circle whose sides cut off an arc on the circle that has the same length as the radius of the circle.

  •To convert from degrees to radians, multiply the number of degrees by .

  •To convert from radians to degrees, multiply the number of radians by .

  EXAMPLES

  To convert 60° to radian measure, multiply it by :

  To convert radians to degrees, multiply it by :

  CONVERSIONS WORTH REMEMBERING

  • , and multiples such as .

  • , and 360° = 2π.

  Lesson 9-2: Right-Triangle Trigonometry

  KEY IDEAS

  The Pythagorean theorem relates the measures of the three sides of a right triangle. A trigonometric ratio relates the measures of two sides and one of the acute angles of a right triangle.

  THE THREE BASIC TRIGONOMETRIC RATIOS

  In Figure 9.2 , ΔABC, ΔADE, and ΔAFG each contain a right angle and they have ∠A in common.

  When the ratio of the length of the leg opposite ∠A to the length of the hypotenuse is computed, it is found to be the same for each right triangle:

  This constant ratio is called the sine of ∠A. By means of a similar approach, the cosine and tangent

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  eBook - ePub

  Construction Mathematics

  • Surinder Virdi, Roy Baker, Narinder Kaur Virdi(Authors)
  • 2014(Publication Date)
  • Routledge

    (Publisher)

  CHAPTER 10

  Geometry

  Learning outcomes: (a) Identify the different types of angles, triangles and quadrilaterals (b) Find angles in triangles, quadrilaterals and other geometrical constructions (c) Use Pythagoras’ theorem to determine diagonals in quadrilaterals and sides of right-angled triangles (d) Calculate the circumference of a circle

  10.1 Angles

  When two straight lines meet at a point an angle is formed, as shown in Figure 10.1 . There are two ways in which an angle can be denoted, i.e. either ∠CAB or ∠A .

  Figure 10.1

  The size of an angle depends on the amount of rotation between two straight lines, as illustrated in Figure 10.2 . Angles are usually measured in degrees, but they can also be measured in radians. A degree, defined as of a complete revolution, is easier to understand and use as compared to the radian. Figure 10.2 shows that the rotation of line AB makes (a) revolution or 90 , (b) revolution or 180 , (c) revolution or270° and (d) a complete revolution or 360°.

  Figure 10.2

  For accurate measurement of an angle a degree is further divided into minutes and seconds. There are 60 minutes in a degree and 60 seconds in a minute. This method is known as the sexagesimal system: 60 minutes (60′) = 1 degree 60 seconds (60″) = 1 minute (1′) The radian is also used as a unit for measuring angles. The following conversion factors may be used to convert degrees into radians and vice versa. 1 radian = 57.30° (correct to 2 d.p.) π radians = 180° (π = 3.14159; correct to 5 d.p.) 2π radians = 360° Example 10.1 Convert: (a) 20°15′25″ into degrees (decimal measure) (b) 32.66° into degrees, minutes and seconds. (c) 60°25′45″ into radians.

  Solution:

  (a) The conversion of 15′25″ into degree involves two steps. The first step is to change 15′25″ into seconds, and the second to convert seconds into a degree. This is added to 20° to get the final answer.

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  eBook - ePub

  Mechanical Engineer's Pocket Book

  • Roger Timings(Author)
  • 2005(Publication Date)
  • Newnes

    (Publisher)

  When a rigid body rotates about a fixed axis all points on the body are constrained to move in a circular path. Therefore in any given period of time all points on the body will complete the same number of revolutions about the axis of rotation; that is, the speed of rotation is referred to in terms of revolutions per minute (rev/min). However, for many engineering problems it is often necessary to express the speed of rotation in terms of the angle in radians turned through in unit time.

  3.7.1 The radian

  A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius of that circle . This is shown in Fig. 3.4(a) . The relationship between arc length, radius and angle in radians is shown in Fig. 3.4(b) .

  Figure 3.4 (a) The radian (r = radius of the circle), (b) Relationship between arc length (s ), radius (r ) and angle (θ ) in radians.

  Where:

  s  = length of arc

  θ  angle in radians

  r  = radius of the circle

  From the definition of a radian:

  s/r = ө therefore s = r θ

  In the expression s = r θ if s is equal to the circumference of a circle then:

  2πr = r θ therefore 2π =ө

  That is, 1 revolution = 2π radians (=360°) Similarly 1 radian = 360/2π = 57.3° (approximately).

  3.7.2 Angular displacement

  In linear motion the symbol s represents the distance travelled in a straight line; that is, the linear displacement. In angular motion, the Greek letter ө is the corresponding symbol for the displacement measured in radians .

  3.7.3 Angular velocity

  A rigid body rotating about a fixed axis O at a uniform speed of n rev/s turns through 2π radians (rad) in each revolution. Therefore the angular velocity ω (Greek letter omega) is given by the expression:

  ω = 2πn rad/s

  Further, if the body rotates through an angle of ө radians in time t seconds then, if that motion is uniform,

  The angular velocity ω = ө /t rad/s

  Note that angular velocity has no linear dimensions.

  3.7.4 The relationship between angular and linear velocity

  Consider the point A on the outer rim of the solid body shown in Fig. 3.5 .

  Figure 3.5

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