Solid Plane Angle and Angle Measure.pptx

For solid and plane geometry

1. Trigonometry and the Unit Circle Math 30-1 1

2. Angles Degrees Standard Position Angle Conversion Radians Coterminal Angles Arc Length A Angles and Angle Measure Math 30-1 2

3. Circular Functions Angles can be measured in: Degrees: common unit used in Geometry Radian: common unit used in Trigonometry Gradient: not common unit, used in surveying Revolutions: angular velocity Math 30-1 3

4. Angles in Standard Position Initial arm Vertex Terminal arm x y To study circular functions, we must consider angles of rotation. Math 30-1 4

5. If the 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 Math 30-1 5

6. Benchmark Angles Special Angles Degrees Math 30-1 6

7. Sketch each rotation angle in standard position. State the quadrant in which the terminal arm lies. 400 ° - 170 ° - 1020 ° 1280 ° Math 30-1 7

8. Coterminal angles are angles 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) = 8

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 9

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) 150 b) -240 c) 570 Positive Negative General Form 510 -210 120 -600 210 -150 Positive Negative General Form Positive Negative General Form , -570 , 480 -510 Math 30-1 10

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. When arc length = radius, the angle measures one radian. How many radians do you think there are in one circle? Math 30-1 11

12. Construct arcs on the circle that are equal in length to the radius. Radian Measure One full revolution is Angles in Standard Position 12

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. 2 r r r Therefore, 2 π rad = 360 . Or, π rad = 180 . r Angle measures without units are considered to be in radians. 13

14. Benchmark Angles Special Angles Radians 14

15. Sketching Angles and Listing Coterminal Angles Sketch the following angles in standard position. Identify all coterminal angles within the domain -4 π < θ < 4 π . Express each angle in general form. a) b) c) Positive Negative General Form Positive Negative General Form Positive Negative General Form 15