FINDING EXACT
TRIGONOMETRIC VALUES
Instructor Brian D. Ray
DRILL
• DIRECTIONS: Solve each special right
triangle shown below.
1)
2)
1
y= 2
S=
t=2
3
45
60
X= 1
1
• In the 45 – 45 – 90 triangle, assume that a leg is 1.
• The other leg is 1 since the 45 – 45 – 90 is isosceles!
• The hypotenuse, by the Pythagorean Theorem is
2 long.
units
DRILL
• DIRECTIONS: Solve each special right
triangle shown below.
1)
2)
1
y= 2
S=
t=2
3
45
x =1
60
1
• In the 30 – 60 – 90 triangle, assume that the short leg is 1.
• How do we know which leg is the short leg?
The short leg is opposite the 30 angle.
•
The hypotenuse is 2 units according to the derivation we did in our previous unit.
• The hypotenuse is 3 units long by the Pythagorean Theorem.
OUR ULTIMATE GOAL
• Why do we learn about functions?
• Do you remember what
kind of function we
used to model each
situation?
Time
(in hrs)
0.5
1
1.5
2
Distance
(miles)
30
60
90
120
OUR ULTIMATE GOAL
• Do you remember what
kind of function we
used to model each
situation?
Path of baseball
Ground zero
OUR ULTIMATE GOAL
• Do you remember what
kind of function we
used to model each
situation?
Verizon charges me $0.45 for
each additional minute that I use
beyond my plan. I used 7:28
additional minutes, but of course,
Verizon will round up, rather than
round down. What function can I
use to model this the additional
cost I would pay?
HERE’S THE POINT
• Have you ever seen this before?
• What about these?
Let’s look here:
http://www.truveo.com/H
ow-to-make-a-y oyosleep-Sleeper-yoyotrick/id/2310084845
• What
function
do we
have to
model
this
motion?
OBJECTIVE
• To model the situations given in the last slides, we
need to learn more trigonometry! Our objective is to
calculate the trigonometric value of any angle,
particularly those having special reference angles.
EXAMPLE
• Find the six trigonometric values for 240 .
Step 1. Draw the angle.
Step 2. Find the reference angle.
Step 3. Set up the special right
triangle. Be careful to use the
correct signs.
Step 4. Apply the definitions we
learned from the reference angle
to find the trigonometric values.
3
opposite   
sin 240 
2
hypotenuse
adjacent   1

cos 240 
2
hypotenuse


90


180


 3



tan 240 
1
360
60
opposite  3
adjacent  1
2

270



opposite
 3

 3
adjacent
1
EXAMPLE
• Find the six trigonometric values for 240 .
Step 1. Draw the angle.
Step 2. Find the reference angle.
Step 3. Set up the special right
triangle. Be careful to use the
correct signs.
Step 4. Apply the definitions we
learned from the reference angle
to find the trigonometric values.
90


180

 3

1
360
60
2
opposite  3
adjacent  1

 
hypotenuse   2   2 3
csc 240 
270 
3
3

opposite

hypotenuse   2  2
adjacent
1
3
sec 240 
cot 240 


1
adjacent
opposite
 3
3


EXAMPLE 2
5

• Find the six trigonometric values for 4 .
Step 1. Draw the angle.
 32   64
Step 2. Find the reference angle.
Step 3. Set up the special right

2
triangle. Be careful to use the
1

8
correct signs.



2

   44 
4
4
Step 4. Apply the definitions we
1
learned from the reference angle
opposite 1

to find the trigonometric values. 
adjacent  1

 


2
opposite  1
 5 

 2   24 
sin  

2

2
 4  hypotenuse
 5
cos 
 4
adjacent

1
2




 hypotenuse
2
2
 5
tan  
 4
1
 opposite



 1
adjacent

1

EXAMPLE 2
5

• Find the six trigonometric values for 4 .
Step 1. Draw the angle.
 32   64
Step 2. Find the reference angle.
Step 3. Set up the special right

2
triangle. Be careful to use the
1

8
correct signs.



2

   44 
4
4
Step 4. Apply the definitions we
1
learned from the reference angle
opposite 1

to find the trigonometric values. 
adjacent  1

 


 5  hypotenuse
2
 2   24 
csc  


 2

opposite
 4 
1
1
 5  hypotenuse  2  2
 5  adjacent
sec  

cot






 1

1
4
adjacent
4
opposite
1




EXAMPLE

330
• Find the six trigonometric values for
.
Step 1. Draw the angle.
Step 2. Find the reference angle.
Step 3. Set up the special right
triangle. Be careful to use the
correct signs.
Step 4. Apply the definitions we
learned from the reference angle
to find the trigonometric values.
opposite   1
sin 330 
2
hypotenuse
3
opposite


cos 330 
2
hypotenuse

90
3

180
opposite
adjacent

tan 330 
30
1

3


2
360
1

270
opposite
1
3


hypotenuse
3
3
EXAMPLE

330
• Find the six trigonometric values for
.
Step 1. Draw the angle.
Step 2. Find the reference angle.
Step 3. Set up the special right
triangle. Be careful to use the
correct signs.
Step 4. Apply the definitions we
learned from the reference angle
to find the trigonometric values.
hypotenuse
csc 330 
opposite


2

 2
1
hypotenuse  2  2 3

sec 330 
3
3
adjacent
90
3

180
opposite
adjacent

cot 330 
30
1

3


360
1
2

270
adjacent
opposite

3
 3
1
Quadrantal Angles
• Definition. A
quadrantile angle is an
angle whose initial side
lies on one of the

coordinates axes.
180

270
90
270

180 
• How do we find trig
values in this case?

360

• Examples.
90
90

360

270
Trigonometric Values of Quadrantal
Angles
90
• Definition. The unit
circle is a circle whose
radius is 1 unit long.
• Identify the ordered

pair for each quadrantal
angle.
• We will now find out
how to find calculate
the trigonometric
values of these angles.
(
0
,
1
)
,

0
(
1
,
0
)
360

180
(
1
1
)

270
(
0
,
1
)
EXAMPLE: Quadrantal Angles
90
• Find the six trigonometric
values for180 .
(
Step 1. Draw the angle.
Step 2. Find the ordered pair from
the unit circle..
Step 3. Apply the definitions we

learned from the reference angle
to find the trigonometric values.
180
y
0

0
r
1
x 1

cos 180  
 1
r
1
y

tan 180   0  0
x 1
sin 180 
(
0
1
,
1
)

0
1
,
0
)
360

,
(
1
)

270
(
0
,
1
)
EXAMPLE: Quadrantal Angles
90
• Find the six trigonometric
values for180 .
(
Step 1. Draw the angle.
Step 2. Find the ordered pair from
the unit circle..
Step 3. Apply the definitions we

learned from the reference angle
to find the trigonometric values.
180
r
0
csc 180 

0
y
1
r

sec 180   1  1
x 1
x
cot 180    1  undefined
y
0
(
0
1
,
1
)


0
1
,
0
)
360

,
(
1
)

270
(
0
,
1
)
Quadrantal Angles
Try This
• Find the six trigonometric
values for  270.
(
Step 1. Draw the angle.
Step 2. Find the ordered pair from
the unit circle..
Step 3. Apply the definitions we

learned from the reference angle
to find the trigonometric values.
180
sin  270  
y 1

r 1 1
x 0

cos 270     0
r 1
y
tan  270    1  undefined
x 0
(
0
1
,
1
)

0
(
1
,
0
)
360

,
90
1
)

270
(
0
,
1
)
Quadrantal Angles
Try This
• Find the six trigonometric
values for 270 .
(
Step 1. Draw the angle.
Step 2. Find the ordered pair from
the unit circle..
Step 3. Apply the definitions we

learned from the reference angle
to find the trigonometric values.
180
r 1
csc  270     1
y 1
r 1

sec  270     undefined
x 0
x 0
cot  270     0
y 1
(
0
1
,
1
)


0
(
1
,
0
)
360

,
90
1
)

270
(
0
,
1
)