4.1 Radian and Degree measure Definition of an angle An angle is made from two rays with a common initial point. Ter min al side Initial side In standard position the initial side is on the x axis Positive angle vs. Negative angle Positive angles are Counter clockwise C.C.W. Negative angles are Clockwise C.W. Angles with the same initial side and terminal side are coterminal. The measure of an angle is from initial side to terminal side Vertex at the origin (Center) Central r Angle r Definition of a Radian Radian is the measure of the arc of a unit circle. Unit circle is a circle with a radius of 1. https://www.youtube.com/watch?v=HhHds http://www.youtube.com/watch?v=7QhgYX 8cRE4 QAM_Rc Radian Protractor Worksheet • Materials – Worksheet – Circle with radius – String – Scissors Complete steps 1-5 The quadrants in terms of Radians What is the circumference of a circle with radius 1? The quadrants in terms of Radians What is the circumference of a circle with radius 1? 2 1 0 2 The quadrants in terms of Radians The circumference can be cut into parts. We’ll start by cutting it into fourths. 2 1 3 2 0 2 The quadrants in terms of Radians I II 2 2 III 3 2 1 3 2 0 0 2 3 2 2 IV 2 • Complete steps 6-8, fill in the blanks • Then starting with a fresh circle on the back of your original circle, write in the radian measurements • Draw a fresh circle in your notes and practice using the patterns of 4ths and 6ths. Radian vs. Degree measurements 360º = 2 180º = So 1 180 rad or 1 rad 180 To convert Degrees into Radians multiply by 180 To convert Radians into Degrees multiply by 180 Change 140º to Radians 140 7 140 * 2.443460953 180 180 9 Change 7 3 to degrees 7 180 1260 * 420 3 3 How to use radians to find Arc length The geometry way was to find the circumference of the circle and multiply by the fraction. Central angle 360º In degrees Arc length called S would be S 360 2r How to use radian to find Arc length In degrees Arc length called S would be S 2r 360 Rewrite the equation replacing 360 degrees with it’s equivalent in radians, then simplify to find the new equation for arc length 2r S r S 2 S r Find the arc length in radians r = 9, θ = 50º Changing to rads 5 50 180 18 r Arc length S 5 S 9 18 5 S 7.85 2 Find the Coterminal Angle Since 2 equals 0. it can be added or subtracted from any angle to find a coterminal angle. Given 3 4 3 5 2 4 4 3 11 2 4 4 Linear speed and Angular speed Linear speed is how fast a particle moves along a circular arc. Angular speed, is how fast the angle changes. • How do we calculate how fast we are driving? • mi/hr or the distance divided by the time • The same idea applies to linear and angular speed Linear speed and Angular speed Linear speed, v = arc length S time Angular speed, Lower case “Omega” t Central angle time t Assuming “constant speed” A Ferris wheel has a 50 ft radius and makes 1.5 revolutions per minute. What is the linear and angular speed in radians? S r (50)(3 ) v 150 471.2 feet per min t t 1 min 3 rad 3 radians t 1 min arclength S v time t Centralangle time t H Dub • 4-1 Pg. 290 #1-10, 15-18, 21, 22, 55-70, and 103