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Section5.1 Angles and Their Measure Angles A ray is a part of a line that has only one endpoint and extends forever in the opposite dire ction. A n angle is form ed by tw o rays that have a com m on endpoint. O ne ray is called the initial side and the other the term inal side. A rotating ray is often a useful w ay to think about angles. T he endpoint of an angle's initial side and term inal side is the vertex of the angle. A n angle is in standard pos ition if 1. its vertex is at the origin of a rectangular coordinate system 2. its initial side lies along the positive x-axis P ositive angles are generated by counterclockw ise rotation. T hus, angle is positive. N egative angles are generated by clockw ise rotation as you see angle in the diagram . A n angle is called quadran tal if its term inal side lies on the x-axis or the y-axis. If a standard ang le has a term inal side that lies in a quadrant then w e say that the angle lies in that quadrant. A ngle lies in quadrant II. A ngle lies in quadrant III. Measuring Angles Using Degrees Names of Angles Degree-Minute-Second • 1 Degree = 60 Minutes • 1 Minute = 60 seconds • Thus »1° = 60’ » 1’ = 60” »1° = 3600” • Convert 50°6’21” to a decimal in degrees • Convert 40°10’25” to a decimal in degrees • Convert 73°40’40” to a decimal in degrees • Convert 21.256° to degree-minute-second • Convert 18.255° to degree-minute-second • Convert 29.411° to degree-minute-second 5 Min Challenge • Convert to a decimal 30 15 '10 " 2 43 '18 " 22 31 '40 " 123 20 '9 " • Convert to D-M-S 30 . 42 31 . 73 51 . 37 127 . 18 Measuring Angles Using Radians Example W hat is the radian m easure of for an ar c of 20 inches length 20 inches and a radius of 5 inche s. 5 inches Relationship between Degrees and Radians Example C onvert each angle in degrees to radians . a. 135 0 b. -120 0 c. -150 0 d. 90 0 e. 180 0 Example C onvert each angle in radians to degrees. a. 2 b. c. 3 d. 5 6 e. 2 3 Top 10 #1-5 • Convert the following to degree measure 9 4 5 3 2 2 3 7 4 Top 10 #6-10 • Convert the following to radian measure • • • • • 220° 315° -90° 900° -270° Drawing Angles in Standard Position Angles Formed by Revolution of Terminal Sides Example D raw and label each angle in standard po sition. a. 3 2 b. = 2 c. = 7 4 Degree and Angle Measures of Selected Positive and Negative Angles Coterminal Angles Example A ssum e the follow ing angles are in stand ard position. 0 F ind a positive angle less than 360 that is coterm inal w ith each of the follow ing. a. 390 b. 405 0 0 c. -135 0 Example A ssum e the follow ing angles are in stand ard position. F ind a positive angle less than 2 that i s coterm inal w ith each of the follow ing. 5 a. 2 b. 1 1 4 c. - 6 Example Find a positive angle less than 2 or 36 0 that is coterm inal 0 w ith each of the follow ing. a. 765 b. 0 22 6 c. - 19 6 Find, and sketch the coterminal angle with… 13 3 2 6 5 Complementary & Supplementary Angles • Complementary angles – Sum of the two angles = 90°; Or… 2 • Supplementary angles – Sum of the two angles = 180° °; Or… Find the complement & Supplement of… 2 4 6 5 5 • Find the complement & supplement of… 3 2 4 8 7 The Length of a Circular Arc Example A circle has a radius of 7 inches. Find the length 0 of the arc intercepted by a central angle of 120 . Example A circle has a radius of 5 inches. Find the length 0 of the arc intercepted by a central angle of 150 . Linear and Angular Speed Example A w indm ill in H olland is used to generate electricity. Its blades are 12 feet in length. T he blades rotate at eight revolutions per m inute. Find th e linear speed, in feet per m inute of the tops of the blades. Exit Slip Convert from degrees to radians 1) 15 2) 120 3) 315 Convert from radians to degrees 4) 5 5) 7 3 5 Draw each angle in standard position 6) 2 7) 5 4 3 8) 8 3 9) 4 Find a positive or negative co-terminal angle with the following 10) 2 11) 13 3 4