Download Section 5.3: Sum and Difference Identities for Cosine Identities

Chapter 5
Section 5.3
Section 5.3: Sum and Difference Identities for Cosine
I.
Identities
Identities



We can use these identities in combination with special angle values to find the exact values for
some “not so special” angles.
Example 1 (Finding Exact Cosine Function Values): Find the exact value of each expression.

a) cos 75

 17 
b) cos 

 12 
Chapter 5
Section 5.3
c) cos173 cos83  sin173 sin83
II.
Cofunction Identities
Identities
Note: We can replace 90 with

.
2
Example 2 (Using Cofunction Identities to Find x): Find one value of x that satisfies each of the
following.
a) sec x  csc62

b) tan x  cot 54

Chapter 5
c) sin

III.
Section 5.3
7
 cos x
6
Note: Because trigonometric (circular) functions are periodic, the solutions in Example 2 are
not unique. We gave only one of infinitely many possibilities.
Applying the Sum and Difference Identities

If either angle A or B in the identities for cos(A + B) and cos(A – B) is a quadrantal angle, then
the identity allows us to write the expression in terms of a single function of A or B.
Example 3 (Rewriting an Expression): Write cos(90° + θ) as a trigonometric function of θ alone.
Chapter 5
Section 5.3
Example 4 (Finding cos (s + t) Given Information About s and t): Suppose that cos s 
15
,
17
sin t  
24
, and both s and t are in quadrant IV. Find cos(s – t).
25
Example 5 (Verify an Identity): Verify that the following equation is an identity.
sec   x    sec x