7-1 7.3 Sum and Difference Identities Cosine Sum and Difference Identities: cos A B does NOT equal cos cos . A B Cosine of a Sum or Difference cos ____________________________________ A B cos ____________________________________ A B EXAMPLE 1 Finding Exact Cosine Function Values Find the exact value of each expression. (a) cos15(b) cos 75° Sine of a Sum or Difference sin ____________________________________ A B sin ____________________________________ A B Tangent of a Sum or Difference tan A B tan A B EXAMPLE 3 Finding Exact Sine and Tangent Function Values Find the exact value of each expression. (a) sin 75° (b) tan 105° (c) sin 40° cos 160° − cos 40° sin 160° (d) cos87 cos93 sin87 sin93
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7-1
7.3 Sum and Difference Identities
Cosine Sum and Difference Identities: cos A B does NOT equal cos cos .A B
Cosine of a Sum or Difference
cos ____________________________________A B
cos ____________________________________A B
EXAMPLE 1 Finding Exact Cosine Function Values Find the exact value of each expression.
(a) cos15 (b) cos 75°
Sine of a Sum or Difference
sin ____________________________________A B
sin ____________________________________A B
Tangent of a Sum or Difference
tan A B
tan A B
EXAMPLE 3 Finding Exact Sine and Tangent Function Values
Find the exact value of each expression.
(a) sin 75° (b) tan 105°
(c) sin 40° cos 160° − cos 40° sin 160° (d) cos87 cos93 sin87 sin93
EXAMPLE 4 Writing Functions as Expressions Involving Functions of
Write each function as an expression involving functions of .
(a) cos 30 (b) tan 45
(c) sin 180
EXAMPLE 5 Finding Function Values and the Quadrant of A + B
Suppose that A and B are angles in standard position, with 4
sin , ,5 2
A A
and 5 3
cos , .13 2
B B
Find each of the following.
(a) sin (A + B) (b) tan (A + B)
7-3
Verifying an Identity
EXAMPLE 7 Verifying an Identity
Verify that the following equation is an identity.
sin cos cos6 3
7.4 Double-Angle and Half-Angle Identities
■ Double-Angle Identities ■ Verifying an Identity
Double-Angle Identities
cos 2A = sin 2A =
cos 2A =
cos 2A = tan 2A =
EXAMPLE 1 Finding Function Values of 2 Given Information about
Given 3
cos5
and sin 0, find sin2 , cos2 , and tan2 .
EXAMPLE 2 Finding Function Values of Given Information about 2
Find the values of the six trigonometric functions of if 4
cos25
and 90 180 .
7-5
EXAMPLE 3 Simplifying Expressions Using Double-Angle Identities
Simplify each expression.
(a) 2 2cos 7 sin 7x x (b) sin15 cos15
EXAMPLE 4 Deriving a Multiple-Angle Identity
Write sin 3x in terms of sin x.
Half-Angle Identities
In the following identities, the symbol _____________ indicates that the sign is chosen based on the function
under consideration and the _____________ of .2
A
cos (𝐴
2) = ___________ sin (
𝐴
2) = ___________ tan (
𝐴
2) = ___________
EXAMPLE 9 Using a Half-Angle Identity to Find an Exact Value
Find the exact value of tan 22.5° using the identity sin