Download Trigonometric Identities

An identity is an equation that is true
for all defined values of a variable.
We are going to use the identities that we have already
established and establish others to "prove" or verify
other identities. Let's summarize the basic identities we
have.
RECIPROCAL IDENTITIES
1
sin x 
csc x
1
cos x 
sec x
1
tan x 
cot x
1
csc x 
sin x
1
sec x 
cos x
1
cot x 
tan x
QUOTIENT IDENTITIES
sin x
tan x 
cos x
cos x
cot x 
sin x
sin x  cos x  1
2
2
Let’s look at the Fundamental Identity derived on page 445
Now to find the two more identities from this famous and oft used one.
sin x  cos x  1
2
2
cos2x
cos2x
What trig function
is this squared?
1
Divide all terms by cos2x
cos2x
What trig function
is this squared?
tan x  1  sec x
2
2
sin x  cos x  1
2
2
sin2x
1
sin2x
Divide all terms by sin2x
sin2x
What trig function What trig function
is this squared?
is this squared?
1  cot x  csc x
2
2
These three are sometimes
called the Pythagorean
Identities since the
derivation of the
fundamental theorem used
the Pythagorean Theorem
RECIPROCAL IDENTITIES
1
sin x 
csc x
1
cos x 
sec x
1
tan x 
cot x
1
csc x 
sin x
1
sec x 
cos x
1
cot x 
tan x
QUOTIENT IDENTITIES
sin x
tan x 
cos x
cos x
cot x 
sin x
PYTHAGOREAN IDENTITIES
1  cot 2 x  csc 2 x
sin 2 x  cos 2 x  1
tan 2 x  1  sec 2 x
All of the identities we learned are found on the back page of your book.
You'll need to have these memorized or be able to derive them for this course.
One way to use identities is to simplify expressions
involving trigonometric functions. Often a good strategy for
doing this is to write all trig functions in terms of sines and
cosines and then simplify. Let’s see an example of this:
substitute using
each identity
sin x
tan x 
cos x
tan x csc x
Simplify:
sec x
simplify
sin x 1

 cos x sin x
1
cos x
1
 cos x
1
cos x
1
1
csc x 
sin x
1
sec x 
cos x
Another way to use identities is to write one function in
terms of another function. Let’s see an example of this:
Write the following expression
in terms of only one trig function:
cos x  sin x  1
2
= 1  sin 2 x  sin x  1
=  sin 2 x  sin x  2
sin 2 x  cos 2 x  1
This expression involves both
sine and cosine. The
Fundamental Identity makes a
connection between sine and
cosine so we can use that and
solve for cosine squared and
substitute.
cos 2 x  1  sin 2 x
A third way to use identities is to find function values. Let’s
see an example of this:
Write the following expression
in terms of only one trig function:
cos x  sin x  1
2
= 1  sin 2 x  sin x  1
=  sin 2 x  sin x  2
sin 2 x  cos 2 x  1
This expression involves both
sine and cosine. The
Fundamental Identity makes a
connection between sine and
cosine so we can use that and
solve for cosine squared and
substitute.
cos 2 x  1  sin 2 x
A third way to use identities is to find function values. Let’s
see an example of this:
1
csc  
sin 
1
Given sin  
with  in quadrant II,
We'd get csc by taking
reciprocal of sin
3
find the other five trig functions using identities.
sin   cos   1
2
2
2
Now use the fundamental trig identity
Sub in the value of sine that you know
1
2
Solve this for cos 
   cos   1
3
8
2 2
8
2
cos   

cos  
3
9
9
square root
both sides
csc  3
When we square root, we need  but determine that we’d
need the negative since we have an angle in Quad II where
cosine values are negative.
1
sin  
3
You can easily find sec by taking reciprocal of cos.
This can be rationalized
3 2
2 2
3
2
sec



csc  3 cos   

2 2 2
3
4
sin 
We need to get tangent using
tan  
fundamental identities.
cos 
This can be rationalized
Simplify by inverting and multiplying
1
1
3


3
1
2
tan  
  




2 2 3  2 2
4
2 2

3
Finally you can find
by taking the
cot   2 2 cotangent
reciprocal of this answer.
Now let’s look at the unit circle to compare trig functions
of positive vs. negative angles.
What is cos
1
2

3
?
 
What is cos   ?
 3
1
2
Remember a negative
angle means to go
clockwise
1
3
 ,

2

2


cos   x   cos x
What is sin

3
Recall from College Algebra that if we put
a negative in the function and get the
original back it is an even function.
?
3
2
 
What is sin    ?
 3
3

2
1
3
 ,

2
2 

sin   x    sin x
What is tan

3
Recall from College Algebra that if we
put a negative in the function and get
the negative of the function back it is an
odd function.
?
3
 
What is tan    ?
 3
 3
1
3
 ,

2
2 

If a function is even, its reciprocal function will be
also. If a function is odd its reciprocal will be also.
EVEN-ODD PROPERTIES
sin(- x ) = - sin x (odd)
csc(- x ) = - csc x (odd)
cos(- x) = cos x (even)
sec(- x ) = sec x (even)
tan(- x) = - tan x (odd)
cot(- x ) = - cot x (odd)
sin  60  what in terms of a positive angle?
 sin 60
 2
sec  
 3

  what in terms of a positive angle?

2
sec
3
RECIPROCAL IDENTITIES
1
csc x 
sin x
1
sec x 
cos x
1
cot x 
tan x
QUOTIENT IDENTITIES
sin x
tan x 
cos x
cot x 
cos x
sin x
PYTHAGOREAN IDENTITIES
2
2
2
2
tan
x

1

sec
x
sin x  cos x  1
1  cot 2 x  csc 2 x
EVEN-ODD IDENTITIES
sin   x    sin x
cos   x   cos x
tan   x    tan x
csc   x    csc x
sec   x   sec x
cot   x    cot x
COFUNCION IDENTITIES
sin(
tan(

2

sec(
2

2
  )  cos 
  )  cot 
  )  csc 
cos(
cot(

2

2
  )  sin 
  )  tan 

csc(   )  sec 
2