Mathematical Investigations IV
Name:
Mathematical Investigations IV
Trigonometry - Beyond the Right Triangles
Sum and Difference Identities
Identities
Identities are statements that are true for all values of the variables.
a b  c  ab  ac


sin     = cos 
2



cos     = sin 
2

a c
ad  bc
 =
b d
bd
sin(–x) = -sin(x)
cos(–x) = cos(x)


csc  =
1
sin 
sec  =
1
cos 
cot  =
cos 
sin 
Some identities follow from the definitions of functions. Others need to be verified. One way to
check any of the above identities is to graph each side of the identity and these graphs, obviously,
should be the same. You should do that with any of the above that you don’t remember. Note
that graphing does not verify, or prove, an identity, but it does reinforce it.
Basic Trigonometric Identities
Pythagorean Identities
From the diagram on the right, the coordinates of P in
terms of  are:
P
1

O
A
P=(
,
)
Using these coordinates as lengths, write the
Pythagorean theorem. This gives a fundamental
identity of trigonometry.
[Hint: PA2 + AO2 = PO2]
This may be rewritten in two other ways:
cos2 =
sin2 =
Trig. 8.1
Rev. F11
Mathematical Investigations IV
Name:
From this basic identity, we can derive two more Pythagorean Identities.
Rewrite the first Pythagorean Identity from the previous page.
Divide both sides by cos2 and simplify. This gives another important identity.
A third Pythagorean Identity is derived similarly to the second. Begin again with the first
identity. This time, divide each term instead by sin2. Simplify.
These three identities will be very important to remember. Make sure that they are correct.
The Sum/Difference Formulas:
Let’s explore how to “expand” cos     and sin     .
cos  
Let’s take the point 
 on the unit circle and apply the rotation matrix
 sin  
cos   sin  
R  
 to rotate this point through an angle  about the origin.
 sin  cos  
A  cos     ,sin     
   B(cos  ,sin  )


O
Now the point that you’ve found is the same as point A above. Set the x- and y-coordinates of
these points equal. The result is:
cos     =
sin     =
Trig. 8.2
Rev. F11
Mathematical Investigations IV
Name:
You should have derived the following:
cos      cos   cos     sin   sin   
sin      sin   cos     sin    cos  
We can now derive the Difference Formulas for Sine and Cosine using the sum formulas.
[Remember that the sine is an odd function and the cosine is an even function.]
cos      cos       
Similarly,
sin      sin       
Sum and Difference formulas for Sine and Cosine
cos      cos   cos     sin   sin   
sin      sin   cos     sin    cos  
cos      cos   cos     sin   sin   
sin      sin   cos     sin    cos  
Trig. 8.3
Rev. F11
Mathematical Investigations IV
Name:
It’s time to play with those formulas! Use the sum and difference formula to simplify the
expressions below:


1. sin     =
2



2. cos     =
2

The identities that you have just derived are called co-function identities. Why? (Look this up
or ask you teacher!)
3. cos     =
4. sin     =
5. Let’s explore one of these sums in a little more detail.
3 

a. Graph y1  cos x and y2  cos  x 
 carefully on the axes below.
2 

b. Looking at your graph of y2 , write this as a sine or cosine with the single argument x.
3 

c. Now use the sum formula for cosine to simplify y2  cos  x 
 . Is this what you
2 

expected?
.
Trig. 8.4
Rev. F11
Mathematical Investigations IV
Name:
1

1
and cos 
with 0   ,   , find the exact values of
3
2
4
each trig function in a-d and an approximate value for each angle in e and f.
6. Given that
cos   


a. sin() =

b. sin  =



c. cos    =
d. cos    =
e.  =
f.  =
g. Compute the difference,    . Next, compute cos1  answer to part c  . Explain any
discrepancies.
h. Compute the sum,    . Next, compute cos1  answer to part d  . Explain any
discrepancies.
Trig. 8.5
Rev. F11