1. 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

2. 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

3. 4.1A Angles and Angle Measure
Degrees
Standard Position
Angle Conversion
Radians
Coterminal Angles
Arc Length
Math 30-1

4. 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

5. 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

6. Positive or Negative Rotation Angle Angles in Standard Positionx 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

7. Benchmark Angles
Special Angles
Degrees
Math 30-1

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

9. 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

10. 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

11. 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)
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

12. 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

13. 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
Math 30-1

14. 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

15. Math 30-1

16. Benchmark Angles
Special Angles
Radians
Math 30-1

17. 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

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