Lecture Two Trigonometric

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1 Lecture Two Trigonometric Section.1 Degrees, Radians, Angles and Triangles Basic Terminology Two distinct points determine line AB. Line segment AB: portion of the line between A and B. Ray AB: portion of the line AB starts at A and continues through B, and past B. A B Angles in General An angle is formed by rays with the same end point. The two rays are the sides of the angle. Angle = AOB O is the common endpoint and it is called vertex of the angle Terminal side O Initial side B A An angle is in a Counterclockwise (CCW) direction: positive angle An angle is in a Clockwise (CW) direction: negative angle Type of Angles: Degree 1

2 Complementary angles: + = 90 Supplementary angles: + = 180 Give the complement and the supplement of each angle: a. 40 Complement: = 50 Supplement: = 140 b. 110 Complement: = -0 Supplement: = 70 Degrees, Minutes, Seconds 1: 1 degree : 1 minute : 1 second full Rotation or Revolution = Change 7.5 to degrees and minutes 7.5 = = 7 +.5(60) = = Add and = = Subtract 4 14 and

3 Angles in Standard Position An angle is said to be in standard position if its initial side is along the positive x-axis and its vertex is at the origin. If angle is in standard position and the terminal side of lies in quadrant I, then we say lies in QI QI If the terminal side of an angle in standard position lies along one of the axes (x-axis or y-axis), such as angles with measures 90, 180, 70, then that called a quadrantal angle. Two angles in standard position with the same terminal side are called coterminal angles. 180 Q II (, +) Q III (, ) Q I (+, +) Q IV (+, ) Find all angles that are coterminal with 10. : k Find the angle of least possible positive measure coterminal with an angle of An angle of 908 is coterminal with an angle of 188 3

4 CD players always spin at the same speed. Suppose a Constant Angular Velocity player makes 480 revolutions per minute. What degrees will a point on the edge of a CD spins for seconds? The player revolves 480 times in one minute times per sec In sec, the CD will spin:.8 = 16 times Therefore; CD will revolve Triangles Equilateral All angles always equal to 60& all sides are equals Isosceles: sides and angles are equals Scalene: No equal sides or angles Right: Has a right angle 90. Obtuse: Has an angle more than 90. Acute: All angles are less than 90. 4

5 Radians Degrees - Radians measures one full rotation 1 1rad The measure of in radians is 1 = 1 degree If no unit of angle measure is specified, then the angle is to be measured in radians. Full Rotation : 360 rad 180 rad Converting from Degrees to Radians rad 1 rad 180 Multiply a degree measure by rad 180 and simplify to convert to radians. Convert 45 to radians 4545 rad rad 5

6 Convert -450 to radians rad 5 rad Convert 49.8 to radians rad rad Converting from Radians to Degrees Multiply a radian measure by 180 radian and simplify to convert to degrees. 180 rad rad Convert 1 to degrees 1 rad Convert 4 3 to degrees Convert 4.5 to degrees

7 s Section.1 Degrees, Radians, Angles and Triangles 1. Indicate the angle if it is an acute or obtuse. Then give the complement and the supplement of each angle. a) 10 b) 5 c) 90 d) 10 e) 150. Change to decimal degrees. 3. Convert to decimal degrees. 4. Convert to decimal degrees. 5. Change to decimal degrees to the nearest thousandth. 6. Convert to degrees, minutes, and seconds. 7. Convert to degrees, minutes, and seconds. 8. Convert to degrees, minutes, and seconds. 9. Convert to degrees, minutes, and seconds. 10. Perform each calculation a) b) c) d) Find the angle of least possible positive measure coterminal with an angle of Find the angle of least possible positive measure coterminal with an angle of Find the angle of least possible positive measure coterminal with an angle of A vertical rise of the Forest Double chair lift 1,170 feet and the length of the chair lift as 5,570 feet. To the nearest foot, find the horizontal distance covered by a person riding this lift. 15. A tire is rotating 600 times per minute. Through how many degrees does a point of the edge of the tire move in 1 second? 16. A windmill makes 90 revolutions per minute. How many revolutions does it make per second? 17. Use a calculator to convert 56 0 to radians to the nearest hundredth of a radian. 18. Convert 78.4 to radians 19. Convert 11 6 to degrees 0. Convert 5 to degrees 3 1. Convert to degrees 6. Use the calculator to convert.4 to degree measure to the nearest tenth of a degree. 7

8 Section.1 Degrees, Radians, Angles and Triangles Indicate the angle if it is an acute or obtuse. Then give the complement and the supplement of each angle. a) 10 b) 5 c) 90 d) 10 e) 150 a) Acute; Complement is = 80; Supplement is = 170. b) Acute; Complement is 90 5 = 38; Supplement is = 18. c) Neither (right angle); Complement is = 0; Supplement is = 90. d) Obtuse; Complement is = 30; Supplement is = 60. e) Obtuse; Complement is = 60; Supplement is = 30. Change to decimal degrees = Convert to decimal degrees

9 Convert to decimal degrees Change to decimal degrees to the nearest thousandth Convert to degrees, minutes, and seconds Convert to degrees, minutes, and seconds

10 Convert to degrees, minutes, and seconds Convert to degrees, minutes, and seconds Convert to degrees, minutes, and seconds

11 Perform each calculation a) b) c) d) a) b) c) d) Find the angle of least possible positive measure coterminal with an angle of

12 Find the angle of least possible positive measure coterminal with an angle of Find the angle of least possible positive measure coterminal with an angle of A vertical rise of the Forest Double chair lift 1,170 feet and the length of the chair lift as 5,570 feet. To the nearest foot, find the horizontal distance covered by a person riding this lift. x x x x 5, ft A 5570 x C B 1170 A tire is rotating 600 times per minute. Through how many degrees does a point of the edge of the tire move in 1 second? rev 1min min 60sec 1 de g/ sec rev A windmill makes 90 revolutions per minute. How many revolutions does it make per second? 90 rev 1min 1. 5 / sec min 60sec rev 5

13 Use a calculator to convert 56 0 to radians to the nearest hundredth of a radian rad Convert to radians rad rad Convert 11 6 to degrees r ad Convert 5 to degrees rad

14 Convert to degrees 6 ( 180 rad) Use the calculator to convert.4 to degree measure to the nearest tenth of a degree..4 rad

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