Math12 (week 1)


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Math12 (week 1)

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