Pre-Calculus
6.3: The Addition and Subtraction Formulas
Sum and Difference Identities
In this section we derive formulas that involve trigonometric functions of u + v or
u – v for any real numbers or angles u and v.
________________________________________________________________
Example 1: Using the subtraction formula for cosine
Find the EXACT value of cos 15 by using the fact that
15 = 60 – 45.
________________________________________________________________________________________________
Example 2: Using the addition formula for cosine
7
7  
 
Find the exact value of cos
by using the fact that
12
12 3 4


________________________________________________________________
Cofunctions
We refer to the sine and cosine functions as cofunctions of each other.
Similarly, the tangent and cotangent functions are cofunctions, as are the secant
and cosecant.
With this, we can consider the right triangle from the other, complementary angle!
Cofunction Ratios
Using ratios, we see that
Example 3: Express as a cofunction of a complementary angle.

a) Sin 26˚15’
b) Tan
9

________________________________________________________________________________________________
More Formulas
________________________________________________________________________________________________
Example 4: Express as a trig function of one angle.
a) sin 23˚ cos 46˚ + cos 23˚ sin 46˚
b) cos (-4) sin 2 – sin 2 cos (-4)
______________________________________________________________________________________________
HW 6.3A: Page 443 #’s 1-15 odd
_______________________________________________________________________________________________
Example 5: Using addition formulas to find the quadrant containing an angle
4
12
Suppose sin  =
cos  = 
where  is in quadrant I and  is in quadrant II.
5
13
Find the exact values of sin ( + ) and tan ( + ).


Reduction Formulas
Addition formulas may also be used to derive reduction formulas. Reduction
formulas may be used to change expressions such as
  
  
sin   n  and cos  n  for any integer n
 2 
 2 
to expressions involving only sin  or cos . Similar formulas are true for the
other trigonometric functions.


Instead of deriving general reduction formulas, we shall illustrate two special
cases in the next example.
_________________________________________________________________________________________________
Example 6: Using reduction formulas
Express in terms of a trigonometric function of  alone:
 3 
(a) sin   

2 
(b) cos ( + )

_________________________________________________________________________________________________
HW 6.3B: Page 444 #’s 17-49odd
You should be adding all these formulas/identities to your list to use on quizzes
and tests.