Download Definitions of Trigonometric Functions of Any Angle

CHAPTER 1.4
CHAPTER 1 TRIGONOMETRY
PART 4 – Trigonometric Functions of Any Angle
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.
2
2
 3.0 - Students know the identity cos (x) + sin (x) = 1.
 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 how to evaluate the trigonometric function of any
angle.
 Students will learn the definition of a reference angle and how to find
it.
 Students will learn how to use reference angles when evaluating a
trigonometric function.
 Students will gain further practice evaluating trigonometric functions
with a calculator.
Introduction
In Section 1.3, the definitions of trigonometric functions were restricted to acute angles.
In this section, the definitions are extended to cover any angle. If  is an acute angle, the
definitions here coincide with those given in the previous section.
Definitions of Trigonometric Functions of Any Angle
bg
Let  be an angle in standard position with x , y a point on the terminal side of
 and r  x 2  y 2  0 .
y
sin  
csc 
cos  
sec 
tan  
cot  
x
CHAPTER 1.4
Because r  x 2  y 2 cannot be zero, it follows that the sine and cosine functions are
defined for any real value of  . However if x = ____, the tangent and secant of  are
undefined. For example, the tangent of 900 is __________________________.
Similarly, if y = ____, the cotangent and cosecant of  are
___________________________.
EXAMPLE 1: Evaluating Trigonometric Functions
Let (-3, 4) be a point on the terminal side of  . Find the sine, cosine, and tangent of  .
y
(-3, 4)
x
Referring to the above diagram, you see that x = -3 and y = 4.
r  x 2  y 2 =__________________________ = ___________ = _________
So, you have sin  
y
x
y
=___________, cos  =_______________, and tan   =
r
r
x
________.
The signs of the trigonometric functions in the four quadrants can be determined easily
from the definitions of the functions. For instance, because cos   ______, it follows
that cos  is __________________ wherever ____ > 0, which is in Quadrants ____ and
_____. (Remember, r is always __________________.)
CHAPTER 1.4
y
Quadrant II
Quadrant I
sin: ______
sin: ______
cos: ______
cos: ______
tan : ______
tan : ______
x
Quadrant III
Quadrant IV
sin: ______
sin: ______
cos: ______
cos: ______
tan : ______
tan : ______
EXAMPLE 2: Evaluating Trigonometric Functions
5
Given tan    and cos  0 , find sin  and sec .
4
Note that  lies in Quadrant _______ because that is the only quadrant in which the
tangent is __________________ and the cosine is _________________. Moreover,
using
tan =__________ = _____________
and the fact that y is negative in Quadrant IV, you can let y = __________ and x =
__________. So, r = ______________________ = _________, and you have
sin 
=
sec =
=
Exact value.

Approximate value.
=
Exact value

Approximate value.
CHAPTER 1.4
EXAMPLE 3: Trigonometric Functions of Quadrant Angles
Evaluate the cosine and tangent functions at the four quadrant angles 0,

2
y
0,1
 1,0
1,0
x

0
3
2
0,1
x
r
=
=
tan 0 =
=
x
r
=
=
tan
cos =
x
r
=
=
tan  =
=
=
tan
cos0 =
cos
cos

2
3
x
=
2
r

3
,  , and
.
2
2
y
x
=
=
=
=
y
x
=
=
3
y
=
2
x
=
=

2
=
y
x
CHAPTER 1.4
1.) (6,-14) is the given point on the terminal side of an angle in standard position.
Determine the exact value of the 6 trigonometric functions of the angle.
y
x
CHAPTER 1.4
2.) Find the values of the 6 trigonometric functions of  .
a.) csc   4,cot   0
b.) tan is undefined, csc  1
y
y
x
3.) Evaluate the following trigonometric functions (when the angle ends between
quadrants use the unit circle).


a.) cos 
b.) tan 3
c.) csc
2
2
F
I
G
HJ
K
x
CHAPTER 1.4
1
x in quadrant III. Find the values of the
3
six trigonometric functions of  by finding a point on the line.
4.) The terminal side of  lies on the line y 
y
x
DAY 1
CHAPTER 1.4
Reference Angles
The values of the trigonometric functions of angles greater than 900 (or less than 0 0 ) can
be determined from their values at corresponding acute angles called reference angles.
Definition of Reference Angle
Let  be an angle in standard position. Its reference angle is the acute angle____ formed
by the terminal side of  and the horizontal axis.
Quadrant II
Quadrant III
Quadrant
IV
' =
' =
' =
' =
' =
' =
CHAPTER 1.4
EXAMPLE 4: Finding Reference Angles
Find the reference angle  ' .
a.)   300 0
Because 3000 lies in Quadrant ______, the angle it makes with the x-axis is
y
x
'
=
=
Degrees.
b.)   2.3

 _____________ and   ________________, it follows
2
that it is in Quadrant _____ and its reference angle is
Because 2.3 lies between
y
x
'
=

Radians.
CHAPTER 1.4
c.)   135 0
First, determine that  1350 is co terminal with ___________, which lies in Quadrant
______. So, the reference angle is
y
x
'
=
=
DAY 2
Degrees.
CHAPTER 1.4
Trigonometric Functions of Real Numbers
To see how a reference angle is used to evaluate a trigonometric function, consider the
point x, y  on the terminal side of  .
y
x, y 
opp
x
adj
By the definition, you know that
sin  
and
tan  
For the right triangle with acute angle ____ and sides of lengths ____ and _____, you
have
sin ' 
=
and
tan ' 
=
So, it follows that _________ and ____________ are equal, except possibly in sign. The
same is true for _________ and __________ and for the other four trigonometric
function. In all cases, the sign of the function value can be determined by the quadrant in
which  lies.
CHAPTER 1.4
Evaluating Trigonometric Functions of Any Angle
To find the value of a trigonometric function of any angle  :
1. Determine the function value for the associated reference angle  ' .
2. Depending on the quadrant in which  lies, prefix the appropriate sign to the
function value.
What Mr. Emhof has memorized
Degrees
30 0
45 0
60 0
sin 
Radians

6

4

3
cos
EXAMPLE 5: Using Reference Angles
Evaluate the trigonometric functions.
4
a.) cos
3
y
x
4
lies in Quadrant _______, the reference angle is  ' 
3
___________________ = _________. Moreover, the cosine is _____________________
in Quadrant ______.
4
cos
=
______________________________
3
Because  
=
_________________________
CHAPTER 1.4

b.) tan  2100

y
x
Because  2100  3600  1500 it follows that  2100 is co terminal with the secondquadrant angle ___________ . Therefore, the reference angle is
 '  _____________________ = ________. Finally, because the tangent is negative in
Quadrant II, you have

tan  2100

=
=
c.) csc
11
4
y
x
11
 11 
Because 
is co terminal with the
  2  ______________ it follows that
4
 4 
second-quadrant angle ___________. Therefore, the reference angle is
 '  _____________________ = ________. Finally, because the cosecant is positive in
Quadrant II, you have
CHAPTER 1.4
csc
11
4
=
=
=
=
5.) Evaluate using reference angles.
a.) sin 300 0
d.) tan
7
6

e.) sec  1200
g.) sec 
DAY 3
 5 
c.) cos 

 4 
b.) cot 1350

f.) csc
5
3
CHAPTER 1.4
EXAMPLE 6: Using Trigonometric Identities
Let  be an angle in Quadrant II such that sin  
1
. Find a.) cos  and b.) tan  by
3
using trigonometric identities.
a.) Using the Pythagorean identity __________________________, you obtain
Substitute ____ for sin  .
Because _____________ in Quadrant II, you can use the __________________ root to
obtain
cos  =
=
b.) Using the trigonometric identity tan  = _____________, you obtain
tan  =
=
=
CHAPTER 1.4
You can use a calculator to evaluate trigonometric functions.
EXAMPLE 7: Using a Calculator
Use a calculator to evaluate each trigonometric function.
Function
a.) cot 4100
Mode
Answer
b.) sin  7
c.) sec

9
5.) Find 2 solutions:
i.) 00    3600
ii.) 0    2
a.) sin  
2
2
b.) sec  2
CHAPTER 1.4


7.) Find 2 solutions 00    3600 .
a.) tan   2.8591
b.) csc  1.0038
CHAPTER 1.4
8.) Find all six trigonometric values in the given quadrant.
9
sec    ; Quadrant III.
4
DAY 4