6.1 Angles and Radian Measure Definitions: An angle is formed by rotating a ray m, called the initial side of the angle, around its endpoint until it coincides with a ray n, called the terminal side of the angle. An angle is made of 2 rays extending from a common endpoint, called the vertex. A One of the rays is considered the initial side while the other ray is the terminal side. B C < ABC is made of 2 rays, BA and BC . B is the common endpoint. B is the vertex of <ABC. <ABC can also be named <B (use the vertex to name the angle). Negative angle is formed. Positive angle is formed. An angle formed by one complete rotation is said to have a measure of 360 degrees. An angle whose terminal side is in standard position lies along a coordinate axis, the angle is said to be a quadrantal angle. QII 360 -360 QIII QI QIV A straight angle measures 180, while a right angle measures 90. An acute angle measures between 0 and 90. An obtuse angle measures between 90 and 180. For two angles to be supplementary, one angle is obtuse and the other is acute or both angles are right. For two angles to be complementary, both angles are acute. Example Find the degree measure of an angle with 1/9 rotation, (a) keeping in mind that one complete rotation corresponds to 360. (b) keeping in mind that one complete rotation corresponds to 2 radians. Solution: (a) We are taking 1/9 of a complete rotation. Remembering that the work “of” means to 1 360 360 40 . So a 1/9 rotation is equivalent to multiply, we have 1/9 x 360 = 9 1 9 a 40 angle. (b) 1/9 x 2 rad.= 1 2 rad . 2 rad . 2 radians . So a 1/9 rotation is equivalent to 9 1 9 9 2/9 radians. Example: Find the radian measure of a central angle opposite a 24 centimeter arc in a circle of radius 4 centimeters. Solution: Looking back to page 417, Definition 2, = s/r radians. In this problem, s is the length of the arc opposite central angle and r is the radius of the circle. s 24cm radians rad . 6 radians r 4cm Example: Find the exact measure of each angle. Remember that 180= radians. a. b. c. d. Change 30 to radian form. Change 3/4 rad. to degree form. Change -/3 to degree form. Change 270 to radian form. Solutions: a. 30 rad . 180 30 rad . 30 rad . 30 rad . radian 1 180 1 180 180 6 b. 3 180 3 180 540 rad . 135 4 rad . 4 1 4 c. 180 1 180 180 rad . 60 3 rad . 3 1 3 d. 270 rad . 180 270 rad . 270 rad . 270 3 rad . radians 1 180 1 180 180 2 Example: Name one positive and one negative angle coterminal with 70º. Solution: Recall one complete rotation is 360º. To make an angle more positive, you will add 360º to the existing angle. 70º + 360º = 430º To make an angle more negative, you will subtract 360º from the existing angle. 70º - 360º = -290º NOTE: If by subtracting 360º to the existing angle does not give you a negative angle, you can continue subtracting 360º as many times needed to achieve a negative angle. Similar holds true for obtaining positive angles.