Angles in Standard Position

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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
1
Radians
Coterminal
Angles
Arc Length
Unit Circle
Points on the
Unit Circle
Trig Ratios
Solving
Problems
Solving
Equations
Degrees
Math 30-1
2
Standard
Position
Angles
Degrees
Angle Conversion
Coterminal
Angles
Radians
Arc Length
Math 30-1
3
Circular Functions
Angles can be measured in:
1
Degrees:
common unit used in Geometry
Radian:
common unit used in Trigonometry
Gradient:
part of a circle
360
not common unit, used in surveying
Revolutions:
angular velocity
Math 30-1
1
2
part of a circle
1
part of a circle
400
radians per second
4
To study circular functions, we must consider angles of rotation.
Angles in Standard Position
Terminal
arm
y

Vertex
Initial arm
Math 30-1
x
5
Positive or Negative Rotation Angle
y
A
If the terminal arm
moves counterclockwise, angle A
is positive.
x
y
A
x
If the terminal side
moves clockwise,
angle A is
negative.
McGraw Hill DVD TeacherMath
Resources
4.1_178_IA
30-1
6
Benchmark Angles
Special Angles
Degrees
120
90
60
45
135
30
150
0 3 60
180
330 
210
225
315 
300
240
Math 30-1
270
7
Sketch each rotation angle in standard position.
State the quadrant in which the terminal arm lies.
400°
- 170°
1280°
-1020°
Math 30-1
8
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 50° + (360°)(-1) = -310°
Clockwise
50° + (360°)(-2) = -670°
Math 30-1
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
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) 1500
b) -2400
Positive
5100
Positive
Negative
-2100 , -5700
Negative
General Form
150  360 n , n  N
c) 5700
1200 , 4800
-6000
General Form
 240  360 n , n  N
Math 30-1
Positive
Negative
2100
-1500 -5100
General Form
570  360 n , n  N
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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.
num ber of radians =
 
arc length
radius
a
r
When arc length = radius, the
angle measures one radian.
How many radians do you
think there are in one circle?
Math 30-1
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Radian Measure
Construct arcs on the
circle that are equal in
length to the radius.
C  2 r
arc length  2  (1)
One full revolution is
2   6.283185307...
radians
Math 30-1
http://www.geogebra.org/en/upload/files/ppsb/radian.html
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.
 =
s = r
r
a
r
   2 rad s
r

r
O
2r
r
1 radian
  = 1 rev o lu tio n o f 3 6 0 
r
Therefore, 2π rad = 3600.
Or, π rad = 1800.
Math 30-1
Angle
measures
without
units are
considered
to be in
radians.
14
Math 30-1
15
Benchmark Angles
Special Angles
Radians

1.57

2
3

4

6
3.14

0
2
6.28
3
Math 30-1
2
4.71
16
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.
5
a)
b)
6

4
3
17 
Positive
Negative 
Positive
6
7
,
6
General Form
5
6
 2 n , n  N
19 
6
c) 1 0 .4 7
Negative
2
,
8
3
3
10 

3
General Form

4
 2 n , n  N
Positive
Negative
4.19
 2.1
,  8.38
General Form
10.47  2  n , n  N
3
Math 30-1
17
Angles and Coterminal Angles
Degrees and Radians
Page 175
1, 6, 7, 8, 9, 11a, c, d, e, h
Math 30-1
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