Download chapter 1 trigonometry

CHAPTER 1.2
CHAPTER 1 TRIGONOMETRY
PART 2 – Trigonometric Functions: The Unit Circle
TRIGONOMETRY MATHEMATICS CONTENT STANDARDS:
 2.0 – Students know the definition of sine and cosine as y-and x-coordinates of
points on the unit circle and are familiar with the graphs of the sine and
cosine functions.
 5.0 – Students know the definitions of the tangent and cotangent functions and
can graph them.
 6.0 – Students know the definitions of secant and cosecant functions and
can graph them.
OBJECTIVE(S):
 Students will learn the definition of a unit circle.
 Students will learn the coordinates of the basic angles in the unit
circle.
 Students will learn how to evaluate the six basic trigonometric given
an angle on the unit circle.
 Students will learn the domain and period of sine and cosine.
 Students will learn how to use the period to evaluate the sine and
cosine.
 Students will learn how to evaluate trigonometric functions with a
calculator.
The Unit Circle
The two historical perspectives of trigonometry incorporate different methods for
introducing the trigonometric functions. Our first introduction to these functions is based
on the unit circle.
Consider the unit circle given by: ______________________
Unit Circle
y (0, 1)
x, y 
(-1, 0)
(1, 0) x
x, y 
(0, -1)
CHAPTER 1.2
The Trigonometric Functions
From the preceding discussion, it follows that the coordinates x and y are two functions
of  . You can use these coordinates to define the six trigonometric functions of  .
_____ __________
________
_________
___________
______________
or abbreviated as
_____ __________
________
_________
___________
______________
Definitions of Trigonometric Functions
Let t be a real number or  be an angle and let x, y  be the point on the unit circle
corresponding to t or  .
sin t  ______
cos t  _______
tan t  _______
csc t  ______
sec t  _______
cot t  _______
1. Determine the exact value of the 6 trigonometric functions of the angle  .
y
 4 3
 , 
 5 5
(1, 0)
x
sin  =
csc  =
cos  =
sec  =
tan  =
cot  =
CHAPTER 1.2
There are two unit circles that you are going to have to derive for the rest of the semester
as well as into pre-calculus and calculus. The first unit circle’s angles are multiples of
_____. The other unit circle’s angles are multiples of ______.

Multiples of
4
- The x- and y-coordinate are both _____. The sign of the coordinate depends on the
quadrant.
y
(0, 1)
(_____,______)
(______,_______)
x
(-1, 0)
(1, 0)
(______,______)
(_______,_______)
(0,-1)
Notice how each coordinate makes x 2  y 2  1 true.
CHAPTER 1.2

6
- The x- and y-coordinate are a combination of _____ or ______, but not both. The sign
of the coordinate depends on the quadrant.
Multiples of
-
3
1
_____
2
2
y
(0, 1)
(______,_______)
(______,_______)
(____, ____)
(_____,______)
x
(-1, 0)
(1, 0)
(______,_______)
(____, ____)
(______,______)
(_______,_______)
(0,-1)
Notice how each coordinate makes x 2  y 2  1 true.
CHAPTER 1.2
EXAMPLE 1: Evaluating Trigonometric Functions
Evaluate the six trigonometric functions at each real number.
a.) t 

6
t
sin
cos
tan
b.) t 

6

6

6

6
corresponds to the point (x, y) = (_________, __________)
y
csc
x
sec

y

x
=
=
cot

6

6

6

1

y
=

1

x
=

x

y
=
5
4
5
corresponds to the point (x, y) = (_________, __________)
4
5
5 1
sin
y
csc
 
4
4
y
t
cos
5
x
4
tan
5 y
 
4
x
=
=
sec
5 1
 
4
x
=
cot
5 x
 
4
y
=
c.) t  0
t  0 corresponds to the point (x, y) = (_________, __________)
1
sin 0  y 
csc 0  
y
cos0  x 
tan 0 
y

x
=
=
sec 0 
1

x
=
cot 0 
x

y
=
CHAPTER 1.2
d.) t  
t   corresponds to the point (x, y) = (_________, __________)
1
sin   y 
csc   
y
cos  x 
tan  
y

x
=
=
sec  
1

x
=
cot  
x

y
=
2.) Evaluate, if possible, the six trigonometric functions of the real number t 
EXAMPLE 2: Evaluating Trigonometric Functions

Evaluate the six trigonometric functions at t   .
3
Moving clockwise around the unit circle, it follows that t  

3
3
.
4
, (_______) corresponds
to the point (x, y) = (_____________, _____________).
I
F
G
H3 J
K
F  IJ
cosG
H3 K
F  IJ
tanG
H3 K
I
F
G
H3 J
K
F  IJ
secG
H3 K
F  IJ
cot G
H3 K
csc 
sin 
=
=
=
=
=
CHAPTER 1.2
3.) Evaluate, if possible, the six trigonometric functions of the real number t  

4
.
DAY 1
4.) Evaluate the following trigonometric functions:


a.) sin
b.) cos
6
4

d.) csc
2
e.) sec
2
3

3
f.) cot
3
4
i.) cot
7
6
g.) sin
5
6
h.) cos
j.) csc
5
4
k.) sec
4
3
l.) cot
3
2
n.) cos
7
4
o.) tan
11
6
m.) sin
5
3
p.) sin2
DAY 2
c.) tan
CHAPTER 1.2
Domain and Period of Sine and Cosine
The domain of the sine and cosine functions is the set of all real numbers. To determine
the range of these two functions, consider the unit circle. Because r = 1, it follows that
sin t  y and cost  x . Moreover, because x , y is on the unit circle, you know that
1  y  1and 1  x  1 . So, the values of sine and cosine also range between _______
and _______.
bg
1  y  1
1  x  1
and
1  cost  1
1  sin t  1
Unit circle
y
0,1
 1,0
1,0
x
0,1
Adding 2 to each value of t in the interval 0,2 completes a second revolution around
the unit circle, as shown below. The values of sin t  2 and cos t  2 corresponds to
those of sin t and cost . Similar results can be obtained for repeated revolutions (positive
or negative) on the unit circle. This leads to the general result
b g
b g
cosb
t  2ng
 _______________________
sin t  2n  _______________________
and
b g
CHAPTER 1.2
for any integer n and real number t. Functions that behave in such a repetitive (or cyclic)
manner are called periodic.
t

2
,______________
y
t
t

4
,______________
3
,______________
4
x
t  0 ,____
t   ,______________
t
5
,______________
4
t
t
7
,______________
4
3
,______________
2
Definition of a Periodic Function
A function f is periodic if there exists a positive real number c such that
b g
f t  c  _________
for all t in the domain of f. The smallest number c for which f is periodic is called the
period of f.
CHAPTER 1.2
EXAMPLE 3: Using the Period to Evaluate the Sine and Cosine
13
 ______________________, you have
a.) Because
6
sin
13
6
=
=
=
b.) Because 
7
 ______________________, you have
2
7 I
F
G
H2 J
K
cos 
=
=
=
5.) Evaluate the following trigonometric functions:
31
9
a.) cos
b.) tan 12
c.) sec
6
4
b g
DAY 3
d.) sin
17
3
CHAPTER 1.2
Negative Angle Identities
bg = __________________________
=
__________________________
cosbg
t
=
__________________________
tanbg
t
=
__________________________
cscbg
t
=
__________________________
secbg
t
=
__________________________
cotbg
t
F 3 IJ2 ways.
6.) Evaluate sinG
H4 K
sin t
7.) Use the value of the trigonometric function to evaluate each function.
3
3
a.) sin  t 
b.) cos t  
8
4
bg
bg
bg =
=
secbg
t
i.) sin t
=
i.) cos t
ii.) csct
=
ii.)
Evaluating Trigonometric Functions with a Calculator
When evaluating a trigonometric Function with a calculator, you need to set the
calculator to the desired mode of measurement (degrees or radians).
CHAPTER 1.2
Most calculators do not have keys for the cosecant, secant, and cotangent functions. To
evaluate these functions, you use the _____ key with their respective reciprocal functions

sine, cosine, and tangent. For example, to evaluate csc , use the fact that
8
csc

8
=
__________________________________
EXAMPLE 4: Using a Calculator
2
a.) sin
3
MODE: Radian
8.) Use a calculator to evaluate (RADIAN MODE)

a.) sin
b.) tan 47.3
6
DAY 4
=
___________
b.) cot 15
.
MODE: Radian
2 I
F
G
H3 J
K
c.) sec 