ANGLES AND CARTESIAN PLANE RELATIONSHIP OF ANGLES A. Complementary Angles - are angles whose sum is exactly 90° or 1/2π rad Example a. 30° and 60° b. 40° and 50° c. 59° and 31° d. 14° and 76° RELATIONSHIP OF ANGLES B. Supplementary Angles - are angles whose sum is exactly 180° or πrad Example a. 30° and 150° b. 58° and 122° c. 91° and 89° d. 104° and 76° ACTIVITY 2A Give the complementary and supplementary angles of each given. a. 35° c. 71° b. 89° d. 7° e. 64° STANDARD POSITION OF AN ANGLE x QII QI QIII QIV y The initial side lies on + x-axis. The vertex is on the origin Positive y-axis QII Negative x-axis QI Positive x-axis QIII QIV Negative y-axis PLOTTING OF AN ANGLE IN A CARTESIAN PLANE 1. Place the vertex on the origin, and initial side on the + x-axis. 2. Rotate the terminal side either clockwise or depending on the magnitude of the given angle. ANGLE MEASUREMENT POSITIONS Measurement Position 1 ° − 89° Quadrant 1 91 ° − 179° 181 ° − 269° 271 ° − 359° Quadrant 1I Quadrant 1II Quadrant 1V ANGLESPECIAL MEASUREMENT POSITIONS CASES Angle Position π = 90° Positive y - axis π = 180° Negative x - axis π = 270° Negative y - axis π = 0° & 360° Positive x - axis EXAMPLES 1. π΄° = 150° 150° Location: Quadrant 2 EXAMPLES 2. π½ = 3π πππ π½ = 540° π½ = 180° Location: negative x - axis 180° EXAMPLES 3. ∠πΆ = −140° Location: Quadrant III −140° EXAMPLES 4. π = π 2 πππ π = 90° Location: positive y-axis 90° ACTIVITY 2B Find the position of the angles a. π° = 270° b. πΌ = 7π − 4 πππ c. ∠πΆ = 2ππππ d. π = −60° e. ∠π΄ππΆ = 600° DISTRIBUTION OF ANGLES IN DEGREE AND RADIAN UNITS IN THE CARTESIAN PLANE RADIAN and DEGREE UNITS REFERENCE ANGLE Reference Angle -is any smallest positive acute angle that is measured from the terminal side of a given angle to the nearest x – axis. This angle is denoted by ππ . There are 4 cases that you can choose from. CASE 1 If the position of the angle is on Quadrant 1 ππ = π π = 1 ° − 89° CASE I1 If the position of the angle is on Quadrant 1I ππ = 180° − π π = 91 ° − 179° CASE III If the position of the angle is on Quadrant III ππ = π − 180° π = 181 ° − 269° CASE IV If the position of the angle is on Quadrant IV ππ = 360° − π π = 271 ° − 359° SPECIAL CASES Angle Position Reference Angle π = 90° Positive y - axis ππ = 0 π = 180° Negative x - axis ππ = 0 π = 270° Negative y - axis ππ = 0 π = 0° & 360° Positive x - axis ππ = 0 EXAMPLES π = 91° π = 303° Position: QII Case: 2 ππ = 180° − π ππ = 180° − 91° Position: QIV Case: 4 ππ = 360° − π ππ = 360° − 303° π½π = ππ° π½π = ππ° EXAMPLES π = 408° π = −135° π = 408° − 360° π = 48° Position: QI Case: 1 ππ = π π = 360° − 135° π = 225° π½π = ππ° Position: QIII Case: 3 ππ = π − 180° ππ = 225° − 180° π½π = ππ° PAIR ACTIVITY Positions and Reference Angles (20 pts) CO-TERMINAL ANGLES •Co-terminal angles are two or more different angles that are having the same terminal and the same initial side. CO-TERMINAL ANGLES For counter-clockwise Co-terminal angles = π + 360°π For clockwise Co-terminal angles = π − 360°π Where n is the number of rotation EXAMPLE π = 127° Find the co-terminals by 2 rotations. For counter-clockwise = π + 360°π = 127° + 360°(2) = 127° + 720° = 847° EXAMPLE π = 127° Find the co-terminals by 2 rotations. For clockwise = π − 360°π = 127° − 360°(2) = 127° − 720° = −593° EXAMPLE π = −80° Find the co-terminals by 5 rotations. For counter-clockwise = π + 360°π = −80° + 360°(5) = −80° + 1800° = 1720° EXAMPLE π = −80° Find the co-terminals by 5 rotations. For clockwise = π − 360°π = −80° − 360°(5) = −80° − 1800° = −1880° ACTIVITY 2C Find the co-terminals of the given angle (clockwise and counter-clockwise) with 3 rotations. a. −235° c. 171° b. 109° d. −7° e. 304°
