Standard-Position-of-an-Angle-FULL.ppt

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1. Trigonometry
The study of triangles and the
relationships between their sides
and angles

2. Standard Position
of an Angle

3. Angle in Standard Position
It is easy to see the direction of an angle if it
is presented in reference to a rectangular
coordinate system. Let the origin of the
coordinate system be at the vertex of the
angle and let the positive half of the x-axis
form the initial side of the angle. The
terminal side shall be in any of the four
quadrants of the coordinate system. Angle
drawn in this manner are called angles in
standard position.

4. Angle in Standard
Position
An angle in standard position
has for its initial side the
positive half of the x-axis in a
rectangular coordinate
system.

5. Let’s look at an angle in standard position, where the
initial side is ALWAYS on the positive x-axis and the
vertex is at the origin. The terminal side can be
anywhere and defines the angle.
A positive angle is described
by starting at the initial side
and rotating
counterclockwiseto the
terminal side (angle ).
terminal
side
initial
side
vertex


A negative angle is
described by rotating
clockwise(angle ).

6. Depending upon the degree measure of the angle, the
terminal side can land in one of the four quadrants.
Angles can be larger than 360º
by simply wrapping around the
quadrants again.
(450º, 540º, 630º, 720º, etc.)
III
III IV
III
III IV
-90
-180
-270
-360
III
III
90
180
270
360

7. Name the quadrant of the terminal side.
1)140
o
7) 80
o
2)315
o
8) -475
o
3)-168
o
9) -25
o
4)475
o
10) 1030
o
5)-340
o
11) -1030
o
6)670
o
12) -225
o

8. 

Angles and are
coterminal since they share
the same sides.
Coterminal Anglesare angles that share the same
terminal side, but have different angle measures.
There are also several other
angles that are coterminal to .

9. To find a coterminal angle:
add or subtract 360º (or any multiple of 360
o
)
to the given angle .
Both are
coterminal
angles to 
Example:
35 +360 = 395º
= 35º
35 –360 = -325º
Find a negative and positive coterminal angle to -425
o

10. Find one positive and one negative
coterminal angle for each angle below.
1)140
o
7) 80
o
2)315
o
8) -475
o
3)-168
o
9) -25
o
4)475
o
10) 1030
o
5)-340
o
11) -1030
o
6)670
o
12) -225
o

11. 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.

12. Angles
Degrees
Standard
Position
Angle
Conversion
Radians
Coterminal
Angles
Arc Length

13. Angles in Standard Position
Initial armVertex
Terminal
arm
x
y
To study circular functions, we must consider angles of rotation.

14. 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

15. 30 60 120 150 210 240 300 330 90 180 270 0 Benchmark Angles
Special Angles
Degrees45 135 225 315 360
Math 30-1 15

16. Sketch each rotation angle in standard position.
State the quadrant in which the terminal arm lies.
400° -170°
-1020°1280°
Math 30-1 16

17. Coterminal anglesare 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) =
Negative Coterminal Angles
Clockwise
-310°
770°
-670°
410°
50°+ (360°)(2) =
50°+ (360°)(-1) =
50°+ (360°)(-2) =

18. 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 nis any natural number
Why must n be a natural number?

19. 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
0
b)-240
0
c)570
0
Positive
Negative
General Form
510
0
-210
0
120
0
-600
0
210
0
-150
0150 360 ,n n N 240 360 ,n n N-   570 360 ,n n N
Positive
Negative
General Form
Positive
Negative
General Form
, -570
0
, 480
0
-510
0

20. Radian Measure: Trig and Calculus
The radianmeasure of an angle is the ratioof 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?

21. Construct arcs on the
circle that are equal in
length to the radius.
Radian Measure2 6.283185307...
radians
 C2r arc length2(1)
http://www.geogebra.org/en/upload/files/ppsb/radian.ht
ml
One full revolution is
Math 30-1 21

22. 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, 2πrad = 360
0
.
Or,π rad = 180
0
.
r2 rads
Angle
measures
without
units are
considere
d to be in
radians.
Math 30-1 22

23. Math 30-1 23

24. 6
 3
 2
  3
2
 0 Benchmark Angles
Special Angles
Radians4
 2 1.57 3.14 4.71 6.28
Math 30-1 24

25. 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 Form5
2,
6
n n N

 4
2,
3
n n N

-   10.47 2 ,n n N
Positive
Negative
General Form
Positive
Negative
General Form5
6 4
3

- 10.47 17
6
 7
6

- 19
,
6

- 2
3
 8
,
3
 10
3

- 4.19 2.1- , 8.38-

26. Angles and Coterminal Angles
Degrees and Radians
Page 175
1, 6, 7, 8, 9, 11a, c, d, e, h