Section 5.3
Sum and Difference Identities for Cosine
𝐜𝐨𝐬(𝑨 + 𝑩) = 𝐜𝐨𝐬 𝑨 𝐜𝐨𝐬 𝑩 − 𝐬𝐢𝐧 𝑨 𝐬𝐢𝐧 𝑩
NOTE: Angles A and B can be measured in degrees or radians.
𝐜𝐨𝐬(𝑨 − 𝑩) = 𝐜𝐨𝐬 𝑨 𝐜𝐨𝐬 𝑩 + 𝐬𝐢𝐧 𝑨 𝐬𝐢𝐧 𝑩
We know the exact trigonometric function values of several angles on the unit circle.
0°, 30°, 45°, 60°, 90°, 𝑒𝑡𝑐.
𝜋 𝜋 𝜋 𝜋
6 4 3 2
0, , , , , 𝑒𝑡𝑐. By forming the sums and differences of these angles, we can find exact
function values of angles that are not commonly known.
For example, we know the exact function values for 30° and 45° . Using these identities we can find the exact cosine
value of their sum and of their difference.
Sum
Difference
Example 1: Find the exact value of each expression.
a)
cos 195°
c) cos 173 ° cos 128° + sin 173° sin 128°
𝜋
b) cos (− 12)
d)
cos
𝜋
18
𝜋
𝜋
9
18
cos − sin
sin
𝜋
9
Example 2: Use the identities for the cosine of a sum or difference to write each expression as a function
of 𝜽.
a) cos (𝜃 − 180°)
b) cos (270° + 𝜃 )
Recall the Cofunction Identities from Section 2.1
Degree Form
Radian Form
𝜋
cos(90° − 𝜃 ) = sin 𝜃
sin(90° − 𝜃 ) = cos 𝜃
cos (2 − 𝑥) = sin 𝑥
sec(90° − 𝜃 ) = csc 𝜃
csc(90° − 𝜃 ) = sec 𝜃
sec (2 − 𝑥) = csc 𝑥
tan(90° − 𝜃 ) = cot 𝜃
cot(90° − 𝜃 ) = tan 𝜃
𝜋
𝜋
tan ( 2 − 𝑥) = cot 𝑥
Use the Difference Identity for Cosine to verify these cofunction identities.
Degree Form:
cos(90° − 𝜃 ) = sin 𝜃
Radian Form:
𝜋
cos ( − 𝑥) = sin 𝑥
2
𝜋
sin (2 − 𝑥) = cos 𝑥
𝜋
csc ( 2 − 𝑥) = sec 𝑥
𝜋
cot ( 2 − 𝑥) = tan 𝑥
Example 3: Verify that each equation is an identity.
NOTE: cos 2𝑥 = cos(𝑥 + 𝑥)
2
a) cos 2𝑥 = cos2 𝑥 − sin 𝑥
b)
1 + cos 2𝑥 − cos2 𝑥 = cos2 𝑥