Angles and Their Measure Find Degree Measure An angle is formed by rotating a ray about its endpoint. Initial side of the angle is the starting position of the ray. Terminal side of the angle is the final position of the ray. Vertex of the angle is the common endpoint, often named by a capital letter such as A. Terminal side B A Initial side Vertex An angle is in standard position if its vertex is at the origin in the xy-plane, and its initial side is the positive x-axis. Vertex is at the origin. Initial side is positive x-axis. The measure of an angle quantifies the direction and amount of rotation from the initial side to the terminal side. The measure of an angle is positive if the rotation is counterclockwise. The measure of an angle is negative if the rotation is clockwise. 45 –150 A degree is one way to measure an angle. One full rotation of a ray about its endpoint is 360 degrees, denoted 360. Therefore, 1 is 1. 1 360 of a full rotation. Sketch the angles in standard position: a. 120° b. –70° Named Angles: Right angle Measures 90 (one-quarter turn) Straight angle Measures 180 (one-half turn) 90 Acute angle (0 <  < 90) Obtuse angle (90 <  < 180) 180 Complementary angles: sum of the measures of two angles is 90 Supplementary angles: sum of the measures of two angles is 180 The complement of a 30° angle is a _____° angle. The supplement of a 130° angle is a _____° angle. A degree can be divided into 60 equal parts called minutes (min or ). 1 degree = 60 minutes or 1 60 1 minute = degree Each minute is divided into 60 equal parts called seconds (sec or ). 1 1 minute = 60 seconds, 1 degree = 3600 seconds 2. Convert to decimal degrees: 38 12 3. Convert to degrees, minutes, seconds (DMS): 120.45° or 1 1 second = 60 minute = 3600 degree Find Radian Measure When two lines or rays cross a circle, the part of the circle between the intersection points is called the intercepted arc and is often denoted by s. A central angle is an angle with the vertex at the center of a circle. Length of arc (denoted by s) equals the radius s=r = 1 radian r A central angle that intercepts an arc on the circle with length equal to the radius of the circle has a measure of 1 radian. 1 radian is simply written as 1. Radian measure carries no units. Radian Measure of an Angle: The radian measure of a central angle  subtended by an arc of length s on a   circle of radius r is given by Number of radians in one revolution: s r . s circumference of circle 2 r    2 6.28 radius of circle r r The measure of one full revolution in degrees is 360°. Thus we have 2 360 or  180 .  Converting Between Degrees and Radians: degrees to radians, multiply the degree measure by 180 180 radians to degrees, multiply the radian measure by  4. Convert from degrees to radians. a. 120° 5. Covert from radians to degrees. a. 9 b. –300° 8 b. –1.8 Determine Coterminal Angles Coterminal Angles: Two angles in standard position with the same initial side and same terminal side. Two angles in standard position are coterminal if their measures differ by a multiple of 360  or 2 radians.  = 50° and  = 410° 50° + 360° = 410° 11    and   6 6 11   2  6 6 y y x x 6. Find a positive angle and a negative angle that is coterminal to the given angle.  a. 7. 110° b.  4 Find an angle between 0 and 360 that is coterminal to the given angle. a. 910° b. –80° 8. Find and angle between 0 and 2 that is coterminal to the given angle. a. 17 6  b. 35 12 Compute Arc Length of a Sector of a Circle Arc Length: 9. Given a circle of radius r, the length s of an arc intercepted by a central angle  (in radians) is given by s r . r Find the exact length of the arc intercepted by a central angle  on a circle of radius r. Then round to the nearest tenth of a unit. a.    , r 14 m 12 b.  20 , r 2 ft 10. Key West, Florida is located at 24.1°N, 81.8W, and Cleveland, Ohio is located at 41.5N, 81.7W. Since the longitudes are nearly the same, the cities are roughly due north-south of each other. Using the difference in latitude, approximate the distance between the cities assuming that the radius of the earth is 3960 mi. Round to the nearest mile. s Compute Linear and Angular Speed Angular and Linear Speed: If a point on a circle of radius r moves through an angle of  radians in time t, the angular and linear speeds of the point are angular speed:   t v linear speed: s t v or r t or v r . 11. A wheel with a diameter of 1 foot spins at a rate of 40 rpm. a. What is the angular speed? b. How fast in feet per minute does a point on the edge of the wheel spin? Compute the Area of a Sector of a Circle The area A of a sector of a circle of radius r with central angle  (in radians) is given by 1 A  r 2 2 12. A circle has a radius of 12 feet. Find the area of a sector made by an angle of 135°. Find the exact area of the sector. Then round the result to the nearest tenth of a unit.