Finding Reference Angles
It is necessary to be able to make larger angles smaller.
We do this by finding reference angles:
Determine the reference angle
for 140 degree angle.
Step:
1. Start by drawing the given angle
2. Now, we determine how many
degrees it is until we get to the
x-axis (horizon)
40 
140 
3. This is your reference angle
4. A reference angle is always
positive
40 degrees is the reference
angle for 140 degrees
180  140  40
A reference angle must be
an ACUTE angle!!
Determine the reference angle
for a 240 degree angle.
Determine the reference angle
for a 323 degree angle.
240 
323 
60 
240  180  60
360  323  37
In the first quadrant, the
angles are acute, so no need
to find a reference angle.
37 
Determine the reference angle
for a 470 degree angle.
Determine the reference angle
for a -125 degree angle.
470 
70 
55 
125 
470  360  110
180  110  70
180  125  55
Homework
Page 8 & 9
#16,28
Applications of reference angles:
Express as a function of
a positive acute angle:
tan 100 
80 
100 
Step:
1. Start by drawing the given angle
2. Find the reference angle
3. Rewrite the function using reference
angle
4. Determine SIGN of function in
quadrant where it was drawn
tan 80 
What is tan in quadrant II?
 tan 80 
180  100  80
tan 100    tan 80
S
A
T
C
Express as a function of
a positive acute angle:
Express as a function of
a positive acute angle:
cos 241 
sin 492 
492 
241 
48 
61 
492  360  132
241  180  61
cos 61
 cos 61 
180  132  48
sin 48 
Applications of reference angles:
1. Start by drawing the given angle
Find the exact value of
each expression:
Reference
angle
2. Find the reference angle
3. Rewrite the function using reference
angle
cos 120 
60 
Step:
4. Determine SIGN of function in
quadrant where it was drawn
120 
5. Now find exact value using exact
value chart
180  120  60
cos 60
1
 cos 60  
2
Cos in quad II is negative
0
1
2
Since cos is negative in
3
1
2
quadrant II, then the exact
value is negative as well
0
3
3
2
2
3
2
1
2
2
1
2
0
1
3
undefined
Find the exact value of each expression:
tan 315 
0
1
2
2
2
3
2
1
1
3
2
2
2
1
2
0
0
3
3
1
45 
315 
360  315  45
tan 45 
 tan 45   1
3
undefined
Find the exact value of each expression:
sin 120 
60 
cos 210 
120 
210 
30 
180  120  60
3
sin 60 
2
210  180  30
cos 30 
3
 cos 30  
2
Find the exact value of each expression:
sin 600 
tan 30
600 
 30 
60 
600  360  240
240  180  60
sin 60 
3
 sin 60  
2
tan 30 
3
 tan 30  
3
30 
Exact Values of quadrantal angles:
We are going to graph four points and use
our knowledge of the unit circle to help us!
Pcos , sin  
sin 
0,1
cos 
y-value is sin
1,0
1,0
0,1
Just use these points to find the sin or cos
of any angle falling on that line.
sin 90  1
cos 180   1
sin 360  0
Exact Values of quadrantal angles:
How do we find tan of these angles?
0,1
1,0
Remember:
tan 180 
0
sin 180
 0

cos 180 1
tan 270 
0,1
sin 
cos 
1
sin 90
 undefined

0
cos 90
tan 90 
1,0
tan  
tan 360 
undefined
0
Homework
•Page 8 & 9
#8-14,21,24