Section 5.1
Properties/Identities of Trigonometric Functions
Opposite Angle Identities
The identities shown below are called the opposite-angle identities.
Even Functions
cos( x)  cos x
sec( x)  sec x
Odd Functions
sin( x)   sin x
tan( x)   tan x
For example, let x =
csc( x)   csc x
cot( x)   cot x
 2 2
. The ordered pair on the unit circle for this point is 
,
 .
4
 2 2 
 
 
Let’s check: cos     cos   .
 4
4

 
Find cos   .
4
 
Now find cos    .
 4
 
 
Finally, is cos     cos   ?
 4
4
The others can be checked similarly.
Powers of Trigonometric Functions
n
sin n x   sin x  this also applies to any other trig function.
Example 1: Simplify, then factor.
sin 2   x   cos 2   x 
Section 5.1 – Properties/Identities of Trigonometric Functions
1
Periodicity
What is the period of sine, cosine and their reciprocals?
So, if we start at a point P on the unit circle and travel a distance of 2  units, we arrive back at
the same point P; hence, we have the following identities.
sin( t  2k )  sin( t ) , for all real numbers t and all integers k.
cos(t  2k )  cos(t ) , for all real numbers t and all integers k.
sec(t  2k )  sec(t ) , for all real numbers t and all integers k.
csc(t  2k )  csc(t ) , for all real numbers t and all integers k.
2
 
.
For example, we know that sin   =
2
4
Now find:
k=2


sin   4  
4

Example 2: True or False?
a. cos( x  10 )  cos x
k = -2


sin   4  
4

b. csc( x  7 )  csc x
Section 5.1 – Properties/Identities of Trigonometric Functions
2
What is the period of tangent and cotangent?
So, if we start at a point P on the unit circle and travel a distance of  units, we arrive back at
the same point P; hence, we have the following identities.
tan(t  k )  tan(t ) , for all real numbers t and all integers k.
cot(t  k )  cot(t ) , for all real numbers t and all integers k.
 
For example, we know that tan   = 1.
4
Now find:
k=3


tan   3  
4

Example 3: True or False?
a. tan( x  5 )  tan x
k = -3


tan   3  
4

b. cot( x  2 )  cot x
Section 5.1 – Properties/Identities of Trigonometric Functions
3
Pythagorean Identities
sin 2 x  cos 2 x  1
tan 2 x  1  sec 2 x
cot 2 x  1  csc 2 x
We can come up with the second and third Pythagorean identities by simply dividing the first by
either sin 2 x or cos 2 x and using some of the following identities:
Recall:
1
1
1
csc x 
sec x 
cot x 
sin x
cos x
tan x
tan x 
sin x
cos x
cot x 
cos x
sin x
sin 2 x  cos 2 x  1
sin 2 x  cos 2 x  1
Example 4: Is the following statement true:
Start with the LHS:
 csc x    sec x 
2
1
2
 csc x    sec x 
2
1
2
1
1
1
Section 5.1 – Properties/Identities of Trigonometric Functions
4
Sometimes we are given the value of a trig function, but it’s not one familiar to us from the unit
circle. We can simply use the triangle method to still solve the problems. The triangle method
will be used in the following two examples.
Example 5: Suppose that cos( ) 
3

and that 0    . Find each of the following.
2
5
a. sin( )
Example 6: Suppose that sin( )  
b. tan(  5 )
1
3
and that
   2 . Find each of the following.
9
2
a. cos(  10 )
Try this one: Suppose that tan( ) 
b. cot( )
3
3
and that    
. Find each of the following.
4
2
a. sin( )
Section 5.1 – Properties/Identities of Trigonometric Functions
b. sec(  6 )
5