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4.3 Trigonometry Extended: The Circular
Functions
Find a coterminal angle.
Give at least 3 answers for each
Subtract 360º from
60º :
y
- 300º
Subtracty 360º from -120º :
- 480º
240º
60º
x
420º
x
-120º
Add 360º to 420º :
Add 360º to 240º :
780º
600º
These angles will have the same initial and terminal sides.
Coterminal Angles
An angle of xº is coterminal with angles of
xº + k · 360º
where k is an integer.
Assume the following angles are in standard position. Find a
positive angle that is coterminal with:
a. a 30º angle
b. a -2π/3 angle
a.For a 30º angle, add 360º to find the coterminal angle.
30º + 360º = 390º
A 390º angle is coterminal with a 30º angle.
b. For a -2π/3 angle, add 2π (same as 360º) to find the coterminal
angle.
- 2π/3 + 2π = 4π/3
A 4π/3 angle is coterminal with a -2π/3 angle.
y
Text Example
Let P = (-3, -4) be a point on
the terminal side of . Find
each of the six trigonometric
functions of .
5

-3
-5
5
-4
x
2
2

4

3
5
r
-5
P =(-3, -4)
x = -3
y = -4
y 4 4
x 3
3
y 4
4

  , tan   
sin   
  , cos  
x 3 3
r
5
5
r
5
5
x 3 3
r
5
5
r
5
5
,
cot





csc  
  , sec   
y 4 4
x 3
3
y 4
4
The bottom row shows the
reciprocals of the row above.
Definitions of Trigonometric
Functions of Any Angle
If  is an angle in standard position, and
let P = (x, y) be a point on the terminal side
of . If r = x2 + y2 is the distance from
(0, 0) to (x, y), the six trigonometric
functions of  are defined by the following
ratios.
y
sin  
r
y
tan  
x
x0
r
x
r
csc 
sec 
cot  
y
y
x
y0
x0
y0
x
cos 
r
P = (x, y)
r

x
y
Paper Plate Unit Circle
Evaluate, if possible, the cosine function and the sine
function at the following four quadrantal angles and
place them on your paper plate Unit Circle:
 0 0
1
cos 0   1
1
0
sin 0   0
1
(0,1)
(-1,0)
  180  
1
cos    1
1
0
sin    0
1
  90 

2
0
cos90   0
1
(1,0)
1

1
sin
90

(0,-1)
1
3
  270 
2
3
0
cos

0
2
1
3 1
sin

 1
2
1
Definition of a Reference Angle
Let  be a nonacute angle in standard position
that lies in a quadrant. Its reference angle is the
positive acute angle ´ prime formed by the
terminal side or  and the x-axis.
b
Find the reference angle ,
for:
 =315º
315
Solution:
´ =360º - 315º = 45º
a
a
45
b
P(a, b)
DAY 2
More on the Unit Circle Paper Plate and Reference Angles
Find a reference angle,  , for each of the following
angles (and place them on your Unit circle too):
  210
QIII :
     180
60º
30º
π/4
 210  180
   30
  240
QII : since  is negative:
   240  180
   60
7

4
QIV :
   360  
7
 2 
4
8 7


4
4
 

4
The Signs of the Trigonometric Functions
STUDENTS
Quadrant II
SINE positive
y
ALL
Quadrant I
ALL FUNCTIONS positive
(and cosecant)
x
TAKE
Quadrant III
TANGENT positive
CALCULUS
Quadrant IV
COSINE positive
(and cotangent)
(and secant)
If tan  < 0 and cos  >0, name the quadrant
that  lies. IV
Text Example
Use reference angles to find the exact value of
sin 135°
y
Step 1 Find the reference angle
135º terminates in quadrant II with a
reference angle ´ = 180º – 135º = 45º
135°
45°
x
The function value, sin 45º, for the reference angle is
sin
45º
=
2
2
Step 2 Use the quadrant in which  lies to prefix the
appropriate sign to the function value.
Because the sine is positive in quadrant II, put a + sign
before the function value of the reference angle.
2
2

sin 135 = +sin45 = 
2
2
Use reference angles to find the exact value of the
following trigonometric function (sketch it):
sin 5π/3
in QIV, Sine is
negative in QIV
  sin(2  5 / 3)
  sin  / 3
3

2
60º
30º
-
Use reference angles to find the exact value of the
following trigonometric functions (sketch it).
c. sec (-30º)
a. cos 2π/3
b. tan 225º
in QII, Cosine is in QIII, Tangent is
negative in QII positive in QIII
in QIV, Secant is
positive in QIV
2
  cos( 
)   tan(225 180)
3
 tan 45

  cos
1
3
1

2
  sec  30
2

3
2 3

3

 5
7

2
Unit
12 o 2o 12
o
3

3
90
75
105
o
3
o
60
Circle
120
4
o
4
o
45 
5 135
o
o 6
6 150
30

11
o
o
165
15
12
12
 180o
360o or 0o 0 or 2
o 23
345
13 195o
o 12
o
12
330
11
210
7
o
o
315
6
6 5 225
7

o
o
300
240
o
o
5 4
4 4 255 270o 285
19 3
3
3 17
12
12
2
Find cos θ and tan θ by using the given information
to construct a reference triangle:
sin θ = 3/7 and tan θ < 0
θ is in QII
 2
10,3
3
2 10
cos 
or  0.904
7

7
2 10
tan  
3
2 10
10
10
3 10

or  0.474
2 0
Trigonometry
Extended: The
Circular Function