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Botanical Name Narcissus 'Trigonometry' Plant Common Name Trigonometry Daffodil The flowers of a Trigonometry Daffodil are of almost geometric precision with their repeating patterns. Repeating patterns occur in sound, light, tides, time, and nature. To analyse these repeating, cyclical patterns, we need to study the cyclical functions branch of trigonometry. Math 30-1 1 Radians Coterminal Angles Arc Length Unit Circle Points on the Unit Circle Trig Ratios Solving Problems Solving Equations Degrees Math 30-1 2 Standard Position Angles Degrees Angle Conversion Coterminal Angles Radians Arc Length Math 30-1 3 Circular Functions Angles can be measured in: 1 Degrees: common unit used in Geometry Radian: common unit used in Trigonometry Gradient: part of a circle 360 not common unit, used in surveying Revolutions: angular velocity Math 30-1 1 2 part of a circle 1 part of a circle 400 radians per second 4 To study circular functions, we must consider angles of rotation. Angles in Standard Position Terminal arm y Vertex Initial arm Math 30-1 x 5 Positive or Negative Rotation Angle y A If the terminal arm moves counterclockwise, angle A is positive. x y A x If the terminal side moves clockwise, angle A is negative. McGraw Hill DVD TeacherMath Resources 4.1_178_IA 30-1 6 Benchmark Angles Special Angles Degrees 120 90 60 45 135 30 150 0 3 60 180 330 210 225 315 300 240 Math 30-1 270 7 Sketch each rotation angle in standard position. State the quadrant in which the terminal arm lies. 400° - 170° 1280° -1020° Math 30-1 8 McGraw Hill DVD Teacher Resources 4.1_178_IA Coterminal angles are angles in standard position that share the same terminal arm. They also share the same reference angle. 50° Rotation Angle 50° Terminal arm is in quadrant I Positive Coterminal Angles Counterclockwise 50° + (360°)(1) = 410° 50° + (360°)(2) = 770° Negative Coterminal Angles 50° + (360°)(-1) = -310° Clockwise 50° + (360°)(-2) = -670° Math 30-1 9 Coterminal Angles in General Form By adding or subtracting multiples of one full rotation, you can write an infinite number of angles that are coterminal with any given angle. θ ± (360°)n, where n is any natural number Why must n be a natural number? Math 30-1 10 Sketching Angles and Listing Coterminal Angles Sketch the following angles in standard position. Identify all coterminal angles within the domain -720° < θ < 720° . Express each angle in general form. a) 1500 b) -2400 Positive 5100 Positive Negative -2100 , -5700 Negative General Form 150 360 n , n N c) 5700 1200 , 4800 -6000 General Form 240 360 n , n N Math 30-1 Positive Negative 2100 -1500 -5100 General Form 570 360 n , n N 11 Radian Measure: Trig and Calculus The radian measure of an angle is the ratio of arc length of a sector to the radius of the circle. num ber of radians = arc length radius a r When arc length = radius, the angle measures one radian. How many radians do you think there are in one circle? Math 30-1 12 Radian Measure Construct arcs on the circle that are equal in length to the radius. C 2 r arc length 2 (1) One full revolution is 2 6.283185307... radians Math 30-1 http://www.geogebra.org/en/upload/files/ppsb/radian.html 13 Radian Measure One radian is the measure of the central angle subtended in a circle by an arc of equal length to the radius. = s = r r a r 2 rad s r r O 2r r 1 radian = 1 rev o lu tio n o f 3 6 0 r Therefore, 2π rad = 3600. Or, π rad = 1800. Math 30-1 Angle measures without units are considered to be in radians. 14 Math 30-1 15 Benchmark Angles Special Angles Radians 1.57 2 3 4 6 3.14 0 2 6.28 3 Math 30-1 2 4.71 16 Sketching Angles and Listing Coterminal Angles Sketch the following angles in standard position. Identify all coterminal angles within the domain -4π < θ < 4π . Express each angle in general form. 5 a) b) 6 4 3 17 Positive Negative Positive 6 7 , 6 General Form 5 6 2 n , n N 19 6 c) 1 0 .4 7 Negative 2 , 8 3 3 10 3 General Form 4 2 n , n N Positive Negative 4.19 2.1 , 8.38 General Form 10.47 2 n , n N 3 Math 30-1 17 Angles and Coterminal Angles Degrees and Radians Page 175 1, 6, 7, 8, 9, 11a, c, d, e, h Math 30-1 18