Download Properties of Trigonometric Functions

Sullivan Precalculus: Section 5.3
Properties of the Trig Functions
Objectives of this Section
• Determine the Domain and Range of the Trigonometric
Functions
• Determine the Period of the Trigonometric Functions
• Determine the Signs of the Trigonometric Functions in a
Given Quadrant
• Find the Values of the Trigonometric Functions Utilizing
Fundamental Identities
• Use Even-Odd Properties to Find the Exact Value of the
Trigonometric Functions
The domain of the sine function is the set of
all real numbers.
The domain of the cosine function is the set of
all real numbers.
The domain of the tangent function is the set
of all real numbers except odd multiples of
 
 2 90 .
The domain of the secant function is the set
of all real numbers except odd multiples of
 2 90 .
 
The domain of the cotangent function is the
set of all real numbers except integral
multiples of  180 .
The domain of the cosecant function is the
set of all real numbers except integral
multiples of  180 .
Range of the Trigonometric Functions
Let P = (a, b) be the point on the unit
circle that corresponds to the angle  .
Then, -1 < a < 1 and -1 < b < 1.
sin   b
cos  a
 1  sin   1
 1  cos  1
sin   1
cos  1
1
1
csc  

1
sin 
b
csc    1 or csc   1
1
1
sec  

1
cos 
a
sec    1 or sec   1
   tan   
   cot   
A function f is called periodic if there is a
positive number p such that whenever  is
in the domain of f , so is   p, and
f   p  f  p
If there is a smallest such number p, this
smallest value is called the (fundamental)
period of f.
Periodic Properties
sin  2   sin 
csc  2   csc
cos  2   cos
sec  2   sec
tan     tan 
cot     cot 
Find the exact valu e of
(a) sec 390
7
(b) cot
4


(a) sec 390  sec 30  360




2 3
 sec 30 
3

3
 7 
 3

( b ) cot 
    cot
 1
  cot 
4
 4 
 4

y
a < 0, b > 0, r > 0
a > 0, b > 0, r > 0

r
x
(a, b)
a < 0, b < 0, r > 0
a > 0, b < 0, r > 0
y
II , 
sin  0, csc  0
All others negative
III , 
tan  0, cot   0
All others negative
I (+, +)
All positive
IV , 
cos  0, sec  0
All others negative
x
Reciprocal Identities
1
csc 
sin 
1
sec 
cos
1
cot  
tan 
Quotient Identities
sin 
tan  
cos
cos
cot  
sin 
b a c
2
2
2
2
2
b a
c
 2 2
2
c c
c
c
b
2
2
2
b a
    1
c c
90
a
sin   cos   1
2
2
tan   1  sec 
2
2
1  cot   csc 
2
2
1
Given that cos  and  is an acute angle,
4
find the exact value of each of the remaining
five trigonometric function of  .
sin   cos   1
2
2
2
1
sin      1
4
sin  
1
1  
4
2
2
1
sin   1   
4
2
2
1
15
sin   1  
16
4
cos  1 ; sin   15
4
4
sin 
tan 

cos
15
4  15  4  15
1
4 1
4
1
1
4
4 15
csc 



15
sin
15 15
4
1
1
15 sec  1  1  4
cot  


cos 1
tan
15 15
4
Even-Odd Properties
sin     sin 
csc     csc
cos    cos
sec    sec
tan     tan 
cot     cot 
Find the exact valu e of

(a) sin  30

( a ) sin  30




 
(b) cot   
 4
 sin 30

1
 
2

  
(b) cot  
 1
   cot
4
 4 