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Angles in Standard Position
Using the Cartesian plane, you can find the trigonometric
ratios for angles with measures greater than 900 or less
than 00. Angles on the Cartesian plane are called
rotational angles.
An angle is in standard position when the initial arm is on
the positive x-axis and the vertex is at (0, 0).
Terminal
Arm
Initial Arm
Vertex (0, 0)
Angles in Standard Position
An angle is positive
when the rotation is
counterclockwise.
An angle is negative
when the rotation is
clockwise.
Quadrant
II
Quadrant
I
Quadrant
III
Quadrant
IV
Angles in Standard Position
Principal Angle
is measured from the positive x-axis to
the terminal arm.
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is measured in a counterclockwise
direction, therefore is always positive.
is always less than 3600.
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Reference Angle
is the acute angle between the terminal
arm and the closest x-axis.
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Angles in Standard Position
Reference
Angle
Principal
Angle
Principal
Angle
Principal
Angle
Reference
Angle
Reference
Angle
Terminology
Angles are often denoted by lower case
Greek letters, particularly __ (alpha), __(beta),
__(gamma), __(theta), and __(phi).
A positive angle is an angle which is oriented
counter-clockwise. A negative angle is an
angle which is oriented clockwise.
Measuring Angles
There are two types of units used to measure angles:
•Degrees (with minutes and seconds)
•Radians
•1 radian =
180
•1 degree =


180
Degrees, Minutes and Seconds
One full circle is divided into 360 degrees.
Degree measure is denoted by a small
superscripted circle, e.g. 74o.
Degrees, Minutes and Seconds
One degree is divided into 60 minutes.
Minutes are noted by a prime, e.g. 14’.
One minute is divided into 60 seconds.
Seconds are noted by a double-prime, e.g. 15’’.
Radians
Radians are always written with the pi.
Ex.
11
3
Ex. 2
Remark
Two coterminal angles have measures which
differ by any multiple of 3600. To find
coterminal angles, simply add or subtract any
multiple of 3600
More Terminology
If an angle has measure  , then the
complement of this angle has measure 90o- 
These two angles are called complementary.
If an angle has measure  , then the
supplement of this angle has measure
180o-  These two angles are called
supplementary.
Finding the Reference and Principal Angles
Sketch the following angles and list the
reference and principal angles.
B) -1200
C) 800
D) 2400
Principal
Angle
1200
Principal
Angle
2400
Principal
Angle
800
Principal
Angle
2400
Reference
Angle
Reference
Angle
Reference
Angle
800
Reference
Angle
A) 1200
600
600
600
Finding the Trig Ratios of an Angle in Standard Position
Choose a point (x, y) on the terminal arm and
calculate the primary trig ratios.
P(x, y)
r
y
q
x
r2 = x2 + y2
x2 = r2 - y2
y2 = r2 - x2
y
sin q 
r
x
cos q 
r
y
tan q 
x
Finding the Trig Ratios of an Angle in Standard Position
P(x, y)
r
y
q
x
Note that x
is a negative number
r2 = (x)2 + y2
(x)2 = r2 - y2
y2 = r2 - (x)2
y
sin θ 
r
x
cosθ 
r
y
tan θ 
x
Remember that in
quadrant II, x is negative
so cosine and tangent will
be negative.
Finding the Trig Ratios of an Angle in Standard Position
The point P(3, 4) is on the terminal arm of q 
List the trig ratios and find q 
P(3, 4)
5
4
q
3
r2 = x2 + y2
= 32 + 42
= 9 + 16
= 25
r=5
4
sin q 
5
3
cos q 
5
4
tan q 
3
q = 530
Finding the Trig Ratios of an Angle in Standard Position
The point P(-3, 4) is on the terminal arm of q 
List the trig ratios and find q 
P(-3, 4)
4
5
q
-3
r2 = x2 + y2
= (-3)2 + (4)2
= 9 + 16
= 25
r=5
4
sin q 
5
-4
tan q 
3
-3
cos q 
5
q ref= 530
Reference
Angle
Principal Angle
1800 - 530 = 1270
q = 1270
Finding the Trig Ratios of an Angle in Standard Position
The point P(-2, 3) is on the terminal arm of q 
List the trig ratios and find q 
P(-2, 3)
3
13
q
-2
r2 = x2 + y2
= (-2)2 + (3)2
=4+9
= 13
r = √ 13
3
sin q 
13
-3
tan q 
2
-2
cos q 
13
q ref= 560
Reference Angle !!
from your calculator
Principal Angle
1800 - 560 = 1240
q = 1240
Related Angles
Related angles are principal angles that have the
same reference angles. These angles will also have
the same trig ratios. The signs of the ratio may differ
depending on the quadrant that they are in.
PA = 300
300
300
PA = 2100
PA = 1500
300
sin 300 = 0.5
Sin 1500 = 0.5
sin 2100 = -0.5
The Unit Circle
sin
One of the most useful tools in trigonometry is the
unit circle.
It is a circle, with radius 1 unit, that is on the x-y
coordinate plane.
1
cos
The x-axis corresponds to the cosine function, and
the y-axis corresponds to the sine function.
The angles are measured from the positive x-axis
(standard position) counterclockwise.
In order to create the unit circle, we must use the
special right triangles below:
1
30º
30º -60º -90º
45º
1
60º
45º
45º -45º -90º
The hypotenuse for each triangle is 1 unit.
You first need to find the lengths of the other sides of each
right triangle...
45º
1
30º
3
2
60º
1
2
1
45º
2
2
2
2
Now, use the corresponding triangle to find the coordinates on the unit circle...
(0, 1) sin
What are the
coordinates of
This cooresponds
point?
to (costhis
30,sin
30)
 3 1

,  your
(Use
 2 2
30º
(–1, 0)
(0, –1)
3
2
1
2
(cos 30, sin 30)
30-60-90
triangle)
cos
(1, 0)
Now, use the corresponding triangle to find the coordinates on the unit circle...
(0, 1) sin
What are the
(Use your
coordinates of
 245-45-90
2 
this point?

,
(cos45, sin 45)
 2triangle)
2 
 3 1

, 
 2 2
2
2
45º
(–1, 0)
(0, –1)
2
2
(cos 30, sin 30)
cos
(1, 0)
You can use your special right triangles to find any of the points on the unit circle...
(0, 1) sin
 2 2

, 
 2 2 
(cos45, sin 45)
 3 1

, 
 2 2
(cos 30, sin 30)
cos
(1, 0)
(–1, 0)
3
2
1
2
(0, –1)
(Use
Whatyour
are the
 130-60-90
3 
of
 coordinates
, 2triangle)

this2 point?
(cos 270, sin 270)
Use this same technique to complete the unit circle on your own.
(0, 1) sin
 2 2

, 
 2 2 
(cos45, sin 45)
 3 1

, 
 2 2
(cos 30, sin 30)
cos
(1, 0)
(–1, 0)
1
 ,- 3 
2
2 
(0, –1)
(cos 300, sin 300)
Unit πCircle
2
(0, 1)
(-1, 0)π
0 (1, 0)
3π (0, -1)
2
Unit Circle
(0, 1)
π
3 1
( , )
6 2 2
3 1 5π
(, )
2 2 6
30°
(-1, 0)
3 1 7π
(,- )
2
2 6
30°
30
°
30
°
(0, -1)
(1, 0)
11π
3 1
( ,- )
2
2
6
Unit Circle
(0, 1)
1 3 2π
(- ,
)
2 2
3
60°
(-1, 0)
1
3 4π
(- , - )
2 2 3
60°
60
°
60
°
(0, -1)
π 1 3
(
,
)
3 2 2
(1, 0)
5π ( 1 , - 3 )
2 2
3
Unit Circle
(0, 1)
π 2 2
( ,
)
4 2 2
2 2 3π
(,
)
2 2 4
45°
(-1, 0)
2
2 5π
(,)
2
2 4
45°
45
°
45
°
(0, -1)
(1, 0)
7π
2
2
( ,)
2
2
4
Unit Circle
Angles
and
Coordinates
 1 3
- 2, 2 


90°
2
3 3 120
°
5 4 135
6 150 °

2 2
- 2 , 2 



3,1
 2 2



°
180
°7 210
°
3 ,- 1 
6

225
2
2
5
240
°

2
2  4 4 °
- 2 ,- 2 


3 270
 1
3
,
°
 2
2 
(–1, 0)


y (0, 1)


2

1 3
2, 2 


 2 2
 2 , 2 


60°3

4
45° 
30°6
 3 1
 2 ,2


x
0°0
360 2 (1, 0)
330
°
11
315°
6  23 , - 12 


300 ° 7
5
°
3
2
3
(0, –1)
4
 2
2
 2 ,- 2 


1
3
,
2
2 

The Six Trigonometric Functions
Function
Triangle Form
Coordinate Form
Coordinate Form
sin θ
Opposite
Hypotenuse
Vertical Coordinate
Radius
y
r
cos θ
Adjacent
Hypotenuse
Horizontal Coordinate
Radius
x
r
tan θ
Opposite
Adjacent
Vertical Coordinate
Horizontal Coordinate
y
x
cot θ
Adjacent
Opposite
Horizontal Coordinate
Vertical Coordinate
x
y
sec θ
Hypotenuse
Adjacent
Radius
Horizontal Coordinate
r
x
csc θ
Hypotenuse
Opposite
Radius
VerticalCoordinate
r
y