1. PLANE AND SPHERICAL TRIGONOMETRY Angle Measure, Arc Length, Linear and Angular Velocities Engr. Mark Edison M. Victuelles 10/6/2010 1

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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:

8. 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

9. 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

10. Example: Express each angle measure in degrees: a. b. c. d. e. Engr. Mark Edison M. Victuelles 10/6/2010 10

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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