4.1 Radian and Degree Measure: Day 1. Trignometry is the measurement of triangles.

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1 4.1 Radian and Degree Measure: Day 1 Trignometry is the measurement of triangles. An angle is formed by rotating a half-line called a ray around its endpoint. The initial side of the angle remains fixed. A second ray called the terminal side of the angle starts in the initial side position and rotates around a common endpoint, called the vertex, until it reaches its terminal position. Terminal side Vertex Initial side When a terminal side is rotated counterclockwise, the angle is positive. When a terminal side is rotated clockwise, the angle is negative. Two different angles having the same initial and terminal sides are said to be coterminal. An angle is in Standard Position when its initial side is along the positive x-axis and its vertex at the origin. Sketching angles in standard position: To position an angle in standard position, start at 0. (EAST) To draw a positive angle, move counterclockwise. (UP) To draw a negative angle, move clockwise. (DOWN) EX: Sketch the following angles in standard position. Find two coterminal angles, one positive and one negative. Also state which quadrant its terminal side lies. a.) = 125 b.) = 315 c.) = 212

2 Types of Angles: Acute: Right: Standard Units of Measure: Degree Radian Obtuse: Complete revolution = Straight: 1 = of a complete revolution 360 Complementary: Supplementary: Unit Conversion: Decimal to Degree-Minute-Second Degree-Minute-Second to Decimal Type in # of degrees, then to get degree symbol Type in # of minutes, then Type in # of seconds, then to get minute symbol to get second symbol EX2: a) b) '55"

3 Day 2: Radian-Degree Conversion: Radian: A central angle with arc length equal to 1 radius is called 1 radian Since C 2R (1) radians OR 2 radians, and 180 radians (GSP:Trig/Unit Circle) There are just over 6 radius lengths in a circumference of a circle Use the circles below to label the angles in degrees and in radian measure. Converting between degrees and radians: Multiply by a fraction that is equal to 1. To convert from radians to degrees: multiply by 180 ( rads) (rads) To convert from degrees to radians: multiply by 180 (getting rid of radians) (getting rid of notation) EX 3: Convert from degrees to radians or radians to degrees = radians = 3. 4

4 Sketching angles (radians) in standard position: EX 4 Sketch the following in standard position: Then find 1 positive and 1 negative coterminal angle. Then state the quadrant in which the lies a) 3 b)

5 Day 3: Angles and Arcs: Given an arc RQ of a circle with center P, the angle RPQ is said to be the central angle that is subtended by the arc RQ and vice versa. Proportions relating central angles and arcs: s s 360 C 360 A 2 2R 360 R s = arc length C = circumference = angle in degrees R = radius A = area of a sector EX 5: a) C 72", s 12" find b) 15, s 2.5" find r Using Radian Measures: s Solving for gives you: 360 C s 360 Rewrite C = 2 R C s 360 Use conversion to convert to radians 2R s 2R 180 s s( units ) OR you can solve for any of the variables: s R OR no( units ) for R R( units ) A 1 R 2 2 Formula for AREA of a sector

6 EX 6: a) In a circle of radius 5.0 cm, find the central angle subtended by an arc length of 7 cm. b) In a circle of radius 12 feet, find the length of an arc subtended by a central angle of 2.7 radians. c) In a circle of radius 15 cm, find the area of the circular sector with central angle of 2 radians. 3 d) Find the distance between Atlanta( 33 45' N) and Cincinnati( 39 8' N) assuming that they fall on the same longitudinal line. The radius of the earth is 3964 mi.

7 Day 4: S dis tance Linear Velocity: V T time The distance traveled by a point through an arc along a circular path per unit time. Units you will see: m, sec ft, sec miles, etc. hour #radians Angular Velocity: T time The change in the angle when a point travels along a circular path per unit time. radians rads Units you will see:,, etc. sec min Relationship between linear and angular velocity: V R revs / time 2 (answer will be in rads/time) 1 revolution = 2 Diameter D = 2R So s V or V R AND t V or t R Unit Conversion with Revolutions: 1 Revolution = 2 R radians EX 7: Convert 2.7 revs/min to rads/sec

8 EX 8: a) An wind mill has propeller blades that are 5.0 m long. If the blades are rotating at 8 rad/sec, what is the linear velocity (to the nearest m/s) of a point on the tip of one of the blades? b) A point on the rim of a 6.0 ft diameter wheel is traveling at 75 ft/sec. What is the angular velocity of the wheel in radians per second? c) If a 6 cm shaft is rotating at 4000 rpm, what is the speed of a particle on its surface in cm/min?