Narcissus 'Trigonometry' Plant Common Name Trigonometry Daffodil 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

4. Trigonometry and the Unit Circle Degrees Radians Coterminal Angles Arc Length Unit Circle Points on the Unit Circle Trig Ratios Solving Problems Solving Equations Math 30-1

4.1A Angles and Angle Measure Degrees Standard Position Angle Conversion Radians Coterminal Angles Arc Length Math 30-1

Circular Functions Degrees: Radian: Gradient: Revolutions: 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

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

Positive or Negative Rotation Angle Angles in Standard Position x y If the terminal arm moves counter-clockwise, angle A is positive. A x y If the terminal side moves clockwise, angle A is negative. Angles in Standard Position McGraw Hill DVD Teacher Resources 4.1_178_IA Math 30-1

Benchmark Angles Special Angles Degrees Math 30-1

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

Coterminal Angles 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 Clockwise 50° + (360°)(-1) = -310° 50° + (360°)(-2) = -670° Math 30-1

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

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 c) 5700 Positive Negative General Form 5100 Positive Negative General Form 1200 , 4800 Positive Negative General Form 2100 -6000 -1500 -5100 -2100 , -5700 Math 30-1

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

Angles in Standard Position Radian Measure Construct arcs on the circle that are equal in length to the radius. One full revolution is Angles in Standard Position http://www.geogebra.org/en/upload/files/ppsb/radian.html Math 30-1

Radian Measure One radian is the measure of the central angle subtended in a circle by an arc of equal length to the radius. r 2r r Angle measures without units are considered to be in radians. r Therefore, 2π rad = 3600. Or, π rad = 1800. Math 30-1

Math 30-1

Benchmark Angles Special Angles Radians Math 30-1

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. a) b) c) Positive Negative General Form Positive Negative General Form Positive Negative General Form Math 30-1

Assignment: Angles and Coterminal Angles Degrees and Radians Page 175 1, 6, 7, 8, 9, 11a, c, d, e, h Math 30-1