H. Algebra 2 17.1 Notes 17.1 Angles of Rotation and Radian Measure Date: ___________ Explore 1. Drawing Angles of Rotation and Finding Coterminal Angles In the past, we have used angles primarily in Geometry. In Trigonometry, we express angles as angles of rotation. Angle of Rotation, 휽: an angle formed by the starting and ending positions of a ray that rotates about its endpoint. Standard Position: when the starting position of the ray, or initial side of the angle, is on the positive x-axis and has its endpoint at the origin In geometry, you were accustomed to working with angles having measures between 0° and 180°. In trigonometry, angles can have measures greater than 180° and even less than 0°. To see why, think in terms of revolutions, or complete circular motions. Let θ be an angle of rotation in standard position. Direction of Rotation: The direction of rotation determines whether the angle is positive or negative. Counterclockwise: _________________________ Clockwise: _____________________________ Coterminal Angles: angles that share the Number of Rotations: You can have any same terminal side. number of revolutions of an angle of rotation, so angles can be greater For example: The angles with measures of 257° than 360°. and −103°. For example: The angle below measures 420° because it is formed by more than one revolution. A) Draw an angle of rotation of 310°. In what quadrant is the terminal side of the angle? B) Draw a positive coterminal angle. What is the measure of your angle? C) Draw a negative coterminal angle. What is the measure of your angle? D) Describe a general method for finding the measure of 1 coterminal angles. H. Algebra 2 17.1 Notes Understanding Radian Measure The diagram shows three circles centered at the origin. The arcs that are on the circle between the initial and terminal sides of the 225° central angle are called intercepted arcs. 퐴퐵̂ is on a circle with radius 1 unit. 퐶퐷̂ is on a circle with radius 2 units. 퐸퐹̂ is on a circle with radius 3 units. Notice that the intercepted arcs have different lengths, although they are intercepted by the same central angle of 225°. You will now explore how these arc lengths are related to the angle. A) The angle of rotation is ________ degrees counterclockwise. There are ____________ degrees in a circle. 225° represents _______ of the total number of degrees in a circle. So, the length of each intercepted arc is _______ of the total circumference of the circle that it lies on. B) Complete the table. To find the length of the intercepted arc, use the fraction you found in the previous step. Give all answers in terms of 휋. Radius, Circumference, Length of Ratio of Arc Length to 풔 r C Intercepted Arc, s Radius, 푪 = ퟐ흅풓 풓 1 2 3 푠 1. What do you notice about the ratios in the fourth column? 푟 2. Suppose the central angle was 270° instead of 225°? What would the ratios equal? 2 H. Algebra 2 17.1 Notes Converting between Degree Measure and Radian Measure Learning Target A: I can convert angle measures between degrees and radians. Radian Measure: For a central angle 휃 that intercepts an arc of length s 푠 on a circle with radius r, the radian measure of the angle is the ratio 휃 = 푟 Unit Circle: a circle centered at the origin with radius of 1 unit. Radian Measure in a unit circle: 휃 = ______ How many radians are in a full circle? How many degrees are in a full circle? How do you find the circumference of a circle with radius r? Use this information to find the number of radians in a full circle. Converting Degrees to Radians Converting Radians to Degrees Example 1. Convert each measure from degrees to radians and from radians to degrees. 휋 a) 20° f) 8 4휋 b) 315 ° g) 3 9휋 c) 600° h) 2 7휋 d) -60° i) − 12 13휋 e) -540° j) − 6 3 H. Algebra 2 17.1 Notes A) Which is larger a degree or a radian? B) The unit circle at the right show measures of angles commonly used in trigonometry. Fill in the missing angles. C) Convert −495° to radians. 13휋 D) Convert radians to degrees. 12 Solving a Real-World Problem Involving Arc Length Learning Target B: I can solve a real-world problem using arc length. 푠 As you saw in the first Explain, for a central angle θ in radian measure, 휃 = , where s is the intercepted arc 푟 length. Multiplying both sides of the equation by r gives the arc length formula for a circle: Arc Length: For a circle of radius r, and arc length s intercepted by a central angle 휃 (measured in radians): 풔 = 풓휽 Geography The northeastern corner of Maine is due north of the southern tip of South America in Chile. The difference in latitude between the locations is 103°. Using both degree measure and radian measure, and a north-south circumference of Earth of 24,860 miles, find the distance between the two locations. Reflect 1. Given the measure of two angles of rotation, how can you determine whether they are coterminal without actually drawing the angles? 2. What is the conversion factor to go from degrees to radians? What is the conversion factor to go from radians to degrees? How are the conversion factors related? 3. An angle of rotation in standard position intercepts an arc of length 1 on the unit circle. What is the radian measure of the angle of rotation? 4 .