Your SlideShare is downloading. ×
0
Math12 (week 1)
Math12 (week 1)
Math12 (week 1)
Math12 (week 1)
Math12 (week 1)
Math12 (week 1)
Math12 (week 1)
Math12 (week 1)
Math12 (week 1)
Math12 (week 1)
Math12 (week 1)
Math12 (week 1)
Math12 (week 1)
Math12 (week 1)
Math12 (week 1)
Math12 (week 1)
Math12 (week 1)
Math12 (week 1)
Math12 (week 1)
Math12 (week 1)
Math12 (week 1)
Upcoming SlideShare
Loading in...5
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×
Saving this for later? Get the SlideShare app to save on your phone or tablet. Read anywhere, anytime – even offline.
Text the download link to your phone
Standard text messaging rates apply

Math12 (week 1)

670

Published on

Published in: Education, Technology
0 Comments
1 Like
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total Views
670
On Slideshare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
31
Comments
0
Likes
1
Embeds 0
No embeds

Report content
Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
No notes for slide

Transcript

  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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 />

×