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4.1A Angles and Angle Measure.pptx
- 1. Botanical Name Narcissus 'Trigonometry' PlantCommon 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
- 2. Degrees Radians Coterminal Angles Arc LengthUnit Circle Points on the Unit Circle Trig Ratios Solving Problems Solving Equations Math 30-1 2
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- 4. Circular Functions Angles canbe measured in: Degrees: common unit used in Geometry 1 part of a circle 360 Radian: common unit used in Trigonometry 1 part of a circle 2 Gradient: not common unit, used in surveying 1 part of a circle 400 Revolutions: angular velocity radians per second Math 30-1 4
- 5. Angles in StandardPosition Initial arm Vertex Terminal arm x y To study circular functions, we must consider angles of rotation. Math 30-1 5
- 6. If the terminalarm moves counter- clockwise, angle A is positive. A x y If the terminal side moves clockwise, angle A is negative. A x y Positive or Negative Rotation Angle McGraw Hill DVD Teacher Resources 4.1_178_IA Math 30-1 6
- 7. 30 60 120 150 210 240 300 330 90 180 270 0 Benchmark Angles SpecialAngles Degrees 45 135 225 315 360 Math 30-1 7
- 8. Sketch each rotationangle in standard position. State the quadrant in which the terminal arm lies. 400° - 170° -1020° 1280° Math 30-1 8
- 9. Coterminal angles areangles in standard position that share the same terminal arm. They also share the same reference angle. McGraw Hill DVD Teacher Resources 4.1_178_IA 50° Rotation Angle 50° Terminal arm is in quadrant I Positive Coterminal Angles Counterclockwise 50° + (360°)(1) = Negative Coterminal Angles Clockwise -310° 770° -670° 410° 50° + (360°)(2) = 50° + (360°)(-1) = 50° + (360°)(-2) = Math 30-1 9
- 10. Coterminal Angles inGeneral 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
- 11. Sketching Angles andListing 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 c) 5700 Positive Negative General Form 5100 -2100 1200 -6000 2100 -1500 150 360 , n n N 240 360 , n n N 570 360 , n n N Positive Negative General Form Positive Negative General Form , -5700 , 4800 -5100 Math 30-1 11
- 12. Radian Measure: Trigand Calculus The radian measure of an angle is the ratio of arc length of a sector to the radius of the circle. a r number of radians = arc length radius When arc length = radius, the angle measures one radian. How many radians do you think there are in one circle? Math 30-1 12
- 13. Construct arcs onthe circle that are equal in length to the radius. Radian Measure 2 6.283185307... radians C 2r arc length 2(1) http://www.geogebra.org/en/upload/files/ppsb/radian.html One full revolution is Math 30-1 13
- 14. Radian Measure One radianis the measure of the central angle subtended in a circle by an arc of equal length to the radius. 2r r r = a r O r r s = r 1 radian = 1 revolution of 360 Therefore, 2π rad = 3600. Or, π rad = 1800. r 2 rads Angle measures without units are considered to be in radians. Math 30-1 14
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- 17. Sketching Angles andListing 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. a) b) c) Positive Negative General Form 5 2 , 6 n n N 4 2 , 3 n n N 10.47 2 , n n N Positive Negative General Form Positive Negative General Form 5 6 4 3 10.47 17 6 7 6 19 , 6 2 3 8 , 3 10 3 4.19 2.1 , 8.38 Math 30-1 17