Trig Values of Any Angle
Objective: To define trig values
for any angle; to construct the
“Unit Circle.”
Angles
• Lets look at any point. We can use our knowledge of
the six trig functions to make a general statement.
Angles
• Lets look at any point. We can use our knowledge of
the six trig functions to make a general statement.
x2  y2  r 2
r  x2  y2
sin  
y
r
csc  
r
y
cos  
x
r
sec  
r
x
y
tan  
x
cot  
x
y
Example 1
• Let (-3,4) be a point of the terminal side of . Find
the sin, cos, and tan of .
Example 1
• Let (-3,4) be a point of the terminal side of . Find
the sin, cos, and tan of .
• This means that x = -3, y = 4, and by the
Pythagorean Theorem, r = 5.
sin  
4
5
cos   
csc  
3
5
4
tan   
3
5
4
sec   
5
3
cot   
3
4
You Try
• Find all six trig functions for the following angle. You
need to find the value of x first.
You Try
• Find all six trig functions for the following angle. You
need to find the value of x first.
sin   12
13
5
13
csc   12
cos   
5
13
tan   
12
5
sec    135
cot    125
Sign Charts
• The sign of the sin, cos, and tan functions depends on
which quadrant the angle is in. Since the radius is
always positive, the sign of the x and y coordinates
will determine the sign of the trig function.
Sign Charts
• The sin function is positive where y is positive and
negative where y is negative.
• The cos function is positive where the x is positive
and negative where x is negative.
• The tangent function is positive where x and y are the
same sign and negative where they have opposite
signs.
Sign Charts
• This is the sign chart for the sin, cos, and tan
functions. Memorize these, you will need to know
them at all times.
Quadrant Angles
• The benefit of using a unit circle is that the sin value
is just the y coordinate and the cos is the x
coordinate. This leads us to the following:
sin 0  0
cos 0  1
sin 900  1
cos 2  0
sin 180 0  0
cos   1
sin 2700  1
cos 32  0
The Unit Circle
• We will now look at all angles on what we call the
Unit Circle. This is a circle with a radius of 1. This will
make all of our work much easier.
• Since the radius is 1, the sin just becomes the y
coordinate and the cos is the x coordinate.
Reference Angles
• We are going to use reference angles to find values.
A reference angle is the acute angle  formed by the
terminal side of  and the horizontal axis. In other
words, it is the distance to the x axis.
Reference Angles
• We are going to use reference angles to find values.
A reference angle is the acute angle  formed by the
terminal side of  and the horizontal axis. In other
words, it is the distance to the x axis.
Example 4
• Find the reference angle for .
a)  = 1500
b)  = 2400
c)  = -1350
Example 4
• Find the reference angle for .
a)  = 1500
b)  = 2400
c)  = -1350
a) Distance from the x-axis is 30.
b) Distance from the x-axis is 60.
c) Distance from the x-axis is 45.
Example 4
• Find the reference angle for .
a)  = 3000
a) When the angle is greater than 2700, we are
looking for the distance to 3600, not 1800. The
reference angle for this is 600.
Example 4
Find the reference angle for . We can also do this
in radians. We are looking for the distance from .
a)  = 3/4
b)  = 7/6
c)  = -5/3
•
Example 4
Find the reference angle for . We can also do this
in radians. We are looking for the distance from .
a)  = 3/4
b)  = 7/6
c)  = -5/3
•
a) The distance from  is /4.
b) The distance from  is /6.
c) The distance from  is /3.
Using Reference Angles
• Using the three angles in the first quadrant, we can
find the exact value of several other angles. Lets look
at the reference angle of 300, or /6.
(
30 0 , 6
3
2
, 12 )
Using Reference Angles
• Using the three angles in the first quadrant, we can
find the exact value of several other angles. Lets look
at the reference angle of 300, or /6.
(
(
3
2
3
2
, 12 )
, 12 )
(
3
2
(
, 12 )
3
2
, 12 )
Using Reference Angles
• Using the three angles in the first quadrant, we can
find the exact value of several other angles. Lets look
at the reference angle of 600, or /3.
( 12 , 23 )
60 0 , 3
Using Reference Angles
• Using the three angles in the first quadrant, we can
find the exact value of several other angles. Lets look
at the reference angle of 600, or /3.
( 12 ,
3
2
( 12 ,
3
2
)
( 12 ,
)
( 12 ,
3
2
)
3
2
)
Using Reference Angles
• Using the three angles in the first quadrant, we can
find the exact value of several other angles. Lets look
at the reference angle of 450, or /4.
(
450 , 4
2
2
,
2
2
)
Using Reference Angles
• Using the three angles in the first quadrant, we can
find the exact value of several other angles. Lets look
at the reference angle of 450, or /4.
(
(
2
2
2
2
,
,
2
2
)
2
2
)
(
2
2
,
2
2
(
2
2
,
)
2
2
)
The Unit Circle
• This is the unit circle that we will be using from now
on. You need to memorize this and be able to
recreate it for me on every test.
90 0 , 2
0,1
1
2
0 
3
,
3
2
60 ,
3
2
2
0 
4
45 ,
,
1
2
2
3
2
,
1
2
30 0 , 6
 1,0
1,0
180 0 , 
3600 ,2
2700 , 32
0,1
3
3
Using a Reference Angle
• Find the exact value of the following:
a) cos 4/3
b) tan (-2100)
c) csc 11/6
Using a Reference Angle
• Find the exact value of the following:

1
,
Q
3
(

),

a) cos 4/3
3
2
b) tan (-2100)
c) csc 11/6
• First, we find the reference angle.
• Second, we determine which quadrant the angle is
in. This tells us if the answer is positive or negative.
• Third, we find the exact value.
Using a Reference Angle
• Find the exact value of the following:
a) cos 4/3
b) tan (-2100) 30, Q2(), 33
c) csc 11/6
• First, we find the reference angle.
• Second, we determine which quadrant the angle is
in. This tells us if the answer is positive or negative.
• Third, we find the exact value.
Using a Reference Angle
• Find the exact value of the following:
a) cos 4/3
b) tan (-2100)

c) csc 11/6
6 , Q 4( ), 2
• First, we find the reference angle.
• Second, we determine which quadrant the angle is
in. This tells us if the answer is positive or negative.
• Third, we find the exact value.
Using a Reference Angle
• You Try:
• Find the exact value of:
a) sin 3/4
b) cos 3150
c) tan 5/6
Using a Reference Angle
• You Try:
• Find the exact value of:

a) sin 3/4
, Q2(),
4
b) cos 3150
c) tan 5/6
2
2
450 , Q4(),
6 , Q2(),

2
2
3
3
Homework
• Page 479-480
• 1, 3, 11, 13, 15-23 odd, 29-35 odd,
• 45-57 odd