Download Trigonometric Functions Using the Unit Circle Blank Notes

Trigonometric Functions Using the Unit Circle Video Lecture
Section 6.2
Course Learning Objectives:
Demonstrate an understanding of trigonometric functions and their
applications.
Weekly Learning Objectives:
1) Find the coordinates of a terminal point.
2) Find a reference number for an angle.
3) Find the exact values of the trigonometric functions using a point
on the unit circle.
4) Find the exact values of the trigonometric functions of quadrantal
angles.
5) Find the exact values of the trigonometric functions of π/4 = 45⁰.
6) Find the exact values of the trigonometric functions of π/6 = 30⁰.
7) Find the exact values of the trigonometric functions of π/3 = 60⁰.
8) Find the exact values of the trigonometric functions for integer
multiples of π/4 = 45⁰, π/6 = 30⁰, and π/3 = 60⁰.
9) Use a calculator to approximate the value of a trigonometric
function.
Trigonometric Functions Using the Unit Circle
Consider a circle, radius ", center at the origin.
Equation of circle:
Consider a mapping where we mark off a distance, >, around the
unit circle starting at the point Ð"ß !Ñ and move counterclockwise if
> ! and clockwise if > !Þ The point T Bß C that we arrive at is
called the terminal point.
What is the terminal point of each of the following real numbers?
1
a) > œ
#
$1
#
b)
>œ
c)
>œ1
d)
> œ '1
page 1
Find the coordinates of the terminal point if > œ
1
.
%
Use symmetry of the unit circle to find the terminal point for each of
the following.
&1
a) > œ %
b)
>œ
(1
%
c)
>œ
31
4
page 2
Find the coordinates of the terminal point if > œ
1
Þ
'
Use symmetry of the unit circle to find the terminal point for each of
the following.
(1
a) > œ '
b)
>œ
""1
'
c)
>œ
(1
'
page 3
Use symmetry across the line C œ B and the terminal point for
1
1
>œ
to find the coordinates of the terminal point for > œ Þ
'
$
Use symmetry of the unit circle to find the terminal point for each of
the following.
(1
a) > œ $
b)
>œ
""1
$
c)
>œ
#1
$
As you can see from the above examples the terminal point of any
number relates back to the "corresponding" terminal point in the first
quadrant.
page 4
Definition: Let > be a real number. The reference number >
associated with > is the shortest distance along the unit circle
between the terminal point determined by > and the B-+B3=Þ
> measures the angle between > and the closest part of the
B axis. It should be between 0 and 1# Þ
Find the reference numbers for each of the following:
&1
a) > œ
'
&1
%
b)
>œ
c)
>œ
d)
> œ #
&1
$
To find the terminal point for any number >ß
Ð1Ñ Find the reference number >Þ
Ð2Ñ Find the terminal point U+ß , determined by >.
Ð3Ñ The terminal point determined by > is T „ +ß „ , where the signs are
chosen according to the quadrant in which the terminal point lies.
page 5
It is essential that you memorize the coordinates of the special
values of > given in the following table:
> Terminal point determined by >
!
"ß !
È$ "
1
 # ß #
'
1
%
1
$
1
#
È# È#
 # ß # 
" È$
#ß # 
!ß "
Find the coordinates for each of the following using reference numbers.
a)
>œ
(1
$
b)
>œ
"$1
'
c)
>œ
$"1
$
d)
>œ
($1
$
page 6
We can now define the six trigonometric functions:
Definition of the 6 trigonometric functions:
Let > be any real number and let T ÐBß CÑ be the terminal point on the
unit circle determined by >Þ
sin > œ C
csc > œ
cos > œ B
"
ÐC Á !Ñ
C
sec > œ
tan > œ
"
ÐB Á !Ñ
B
C
ÐB Á !Ñ
B
cot > œ
B
ÐC Á !Ñ
C
Find the value of each of the trigonometric functions at
a)
>œ
&1
'
b)
>œ
1
$
c)
>œ
1
#
page 7
The following table shows some special values of the 6
trigonometric functions. These values need to be memorized.
>
!
sin t
!
1
'
1
%
1
$
1
#
"
#
È#
#
È$
#
"
cos >
"
È$
#
È#
#
"
#
!
tan >
!
È$
$
"
È$
—
csc >
—
#
sec >
"
#È $
È#
È#
#È $
$
"
#
#
—
cot >
—
È$
"
È$
$
!
If T ÐBß CÑ is the terminal point for any number >, then ) is the angle
in standard position measured in radians whose terminal side is the
ray from the origin through T .
Since < œ " on a unit circle, and = œ <) with the arc length = œ >,
then > œ " † ) or > œ )Þ
Therefore,
sin ) œ sin >
csc ) œ csc >
cos ) œ cos >
sec ) œ sec >
tan ) œ tan >
cot ) œ cot >
Since the values of the trig functions of an angle ) are determined
by the coordinates of the terminal point T ÐBß CÑ, the units on ) are
irrelevant. ) can be in degrees or radians.
page 8
Find the exact value of each expression:
cosÐ &%1 Ñ
sin "$&°
tan $"&°
cscÐ '!°Ñ
cotÐ &$1 Ñ
sec #"!°
cosÐ%)°Ñ
cscÐ#"°Ñ
tanÐ &"#1 Ñ
cos Ð"Þ'#Ñ
cot Ð"%Þ#Ñ
secÐ)Þ"%Ñ
page 9