Math12 (week 1)
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Math12 (week 1) Math12 (week 1) Presentation Transcript

  • PLANE AND SPHERICAL TRIGONOMETRY
    Angle Measure, Arc Length, Linear and Angular Velocities
    Engr. Mark Edison M. Victuelles
    10/6/2010
    1
  • sPECIFIC OBJECTIVES:
    At the end of the lesson, the student is expected to :
    Define trigonometry.
    Measure angles in rotations, in degrees and in radians.
    Find the measures of coterminal angles.
    Change from degree measure to radian measure and from radian measure to degree measure.
    Find the length of an arc intercepted by a central angle.
    Solve word problems involving arc length
    Solve word problems involving angular velocity and linear velocity
    Engr. Mark Edison M. Victuelles
    10/6/2010
    2
  • Trigonometry
    Trigonometry is the branch of mathematics that deals with the measurement of triangle
    Engr. Mark Edison M. Victuelles
    Angle
    An angle is defined as the amount of rotation to move a ray from one position to another.
    The original position of the ray is called the initial side of the angle, and the final position of the ray is called the terminal side.
    The point about which the rotation occurs and at which the initial and terminal side of the angle intersect is called the vertex.
    10/6/2010
    3
  • Engr. Mark Edison M. Victuelles
    Angle
    Terminal Side
    Positive Angle
    Vertex
    Initial Side
    Negative Angle
    Note:
    When the vertex of an angle is the origin of the rectangular coordinate system and its initial side coincides with the positive x-axis, the angle is said to be in the standard position.
    10/6/2010
    4
  • Angle Measurements
    Degree
    1 revolution = 360 degrees
    a. Minute = 1/60 of a degree
    b. Second = 1/60 of a minute
    Radian
    1 revolution = 2π radians
    Gradian / Gradient / Grade
    1 revolution = 400grads
    Mil
    1 revolution = 6400mils
    Engr. Mark Edison M. Victuelles
    10/6/2010
    5
  • Conversion (angle measurement)
    Engr. Mark Edison M. Victuelles
    10/6/2010
    6
    θ (degrees)
    θ (radians)
    1 revolution = 360 degrees
    1 revolution = 2π radians
    The ratio of
  • Conversion (angle measurement)
    Engr. Mark Edison M. Victuelles
    10/6/2010
    7
    Degrees to Radians:
    Radians to Grad:
    Radians to Degrees:
    Grad to Radians:
    Degrees to Gradians:
    Gradians to Degrees:
  • Example:
    Convert the following angles measured in degrees, minutes and seconds to angles measured to the nearest hundredth of a degree:
    a. 64°24’ 38”
    b. 228° 23’ 10”
    c. 145° 11’ 56”
    d. 356° 09’ 34”
    Engr. Mark Edison M. Victuelles
    10/6/2010
    8
  • Example:
    Convert the following angles measured in degrees to angles measured to the nearest minute:
    a. 56.39°
    b. 273.8°
    c. 323.28°
    d. 163.18°
    Engr. Mark Edison M. Victuelles
    10/6/2010
    9
  • Example:
    Express each angle measure in degrees:
    a.
    b.
    c.
    d.
    e.
    Engr. Mark Edison M. Victuelles
    10/6/2010
    10
  • Example:
    Express each angle measure in radians. Give answer in terms of :
    a. 120°
    b. 335°
    c. -310°
    d. 1035°
    e. 450°
    Engr. Mark Edison M. Victuelles
    10/6/2010
    11
  • Coterminal Angles
    Coterminalangles are angles in standard position whose initial and terminal sides are the same.
    To find angles coterminal to a given angle, add or subtract multiples of 360° to it.
    Engr. Mark Edison M. Victuelles
    10/6/2010
    12
  • Example:
    Draw the following angles and find two angles (one positive and one negative) coterminal with each.
    a. 55°
    b. 70°
    c. 153°
    d. 219°
    Engr. Mark Edison M. Victuelles
    10/6/2010
    13
  • Example:
    For each of the following angles, find a coterminal angle with measure such that
    a. -100°
    b. 524°
    c. 900°
    d. 1250°
    Engr. Mark Edison M. Victuelles
    10/6/2010
    14
  • Classification of Angles
    Angles are classified according to the measurement of its angle.
    Zero Angle – an angle formed by two coinciding rays without rotation between them
    Acute angle (0r sharp) – an angle formed between 0 and 90.
    Right Angle – is a 90 angle. Angle formed by two perpendicular rays.
    Engr. Mark Edison M. Victuelles
    10/6/2010
    15
  • Classification of Angles
    Obtuse Angle (Blunt) – angle formed between 90 and 180.
    Straight Angle – an angle whose measure is exactly 180 .It is formed by two rays extending in opposite directions.
    Reflex (Bent-Back) – angle formed between 180 and 360.
    Circular Angle – angle whose measure is exactly 360
    Engr. Mark Edison M. Victuelles
    10/6/2010
    16
  • Length of a Circular Arc , s
    Engr. Mark Edison M. Victuelles
    10/6/2010
    17
    An arc length refers to the measure of a position of a circle or part of its circumference. The arc length s for a given central angle can be found as follows:
    where:
    s = length of the arc
    r = radius of the circle
    = measure of the central angle in radians
    s
  • Example:
    Find the length of the arc of a circle whose radius and whose central angle are as follows
    a. = 2.5 radians, r = 20 cm
    b. = 225°, r = 30.1 mm
    c.= , r = 15 ft.
    II. If the minute hand of a clock is 8 cm long, how far does the tip of the hand move after 25 minutes?
    Engr. Mark Edison M. Victuelles
    10/6/2010
    18
    50 cm
    118.2 mm
    82.5 ft
  • Area of a Sector
    Engr. Mark Edison M. Victuelles
    10/6/2010
    19
    where:
    A = area of the sector
    r = radius of the circle
    = measure of the central angle in radians
    A
  • Angular Velocity
    Engr. Mark Edison M. Victuelles
    10/6/2010
    20
    The angular velocity ( ) of a point on a revolving ray is the angular displacement per unit time.
    Where: - is the angular velocity
    - is the angular displacement
    t - time
  • Linear Velocity
    Engr. Mark Edison M. Victuelles
    10/6/2010
    21
    The linear velocity (V) of a point on a revolving ray is the linear distance traveled by the point per unit time.
    Where: V - is the linear velocity
    s - is the linear displacement
    t - time