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Solid Plane Angle and Angle Measure.pptx
- 1. Math 30-1 1 Trigonometryand the Unit Circle Degrees Radians Coterminal Angles Arc Length Unit Circle Points on the Unit Circle Trig Ratios Solving Problems Solving Equations
- 2. Math 30-1 2 Angles Degrees StandardPosition Angle Conversion Radians Coterminal Angles Arc Length A Angles and Angle Measure
- 3. Math 30-1 3 CircularFunctions Angles can be 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
- 4. Math 30-1 4 Anglesin Standard Position Initial arm Vertex Terminal arm x y To study circular functions, we must consider angles of rotation.
- 5. Math 30-1 5 Ifthe terminal arm 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 Angles in Standard Position
- 6. Math 30-1 6 30 60 120 150 210 240300 330 90 180 270 0 Benchmark Angles Special Angles Degrees 45 135 225 315 360
- 7. Math 30-1 7 Sketcheach rotation angle in standard position. State the quadrant in which the terminal arm lies. 400° - 170° -1020° 1280°
- 8. 8 Coterminal angles areangles in standard position that share the same terminal arm. They also share the same reference angle. Coterminal Angles 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) =
- 9. Math 30-1 9 CoterminalAngles 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?
- 10. Math 30-1 10 SketchingAngles 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 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
- 11. Math 30-1 11 RadianMeasure: Trig and 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?
- 12. Construct arcs onthe circle that are equal in length to the radius. Radian Measure 2 6.283185307... radians C 2r arc length 2(1) One full revolution is Angles in Standard Position 12
- 13. 13 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.
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- 15. 15 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