Unit 3B: Trig Identities (Continued)
Essential Questions: How can we find exact values of more angles? How can we use identities to
evaluate more trig expressions?
Rewrite each expression on the left into a sum or difference (of exact value angles) on the right.
tan
17𝜋
12
tan
7𝜋 𝜋
−
4
3
cos 75°
sin
𝜋
12
tan 165°
sin 255°
cos
5𝜋
12
tan 195°
cos
13𝜋
12
sin 345°
tan 285°
Since we were able to write each expression using other angles that we have already worked with, we can apply
SUM and DIFFERENCE IDENTITIES to evaluate the expressions.
SUM
DIFFERENCE
cos 𝐴 + 𝐵 = cos 𝐴 cos 𝐵 − sin 𝐴 sin 𝐵
sin 𝐴 + 𝐵 = sin 𝐴 cos 𝐵 + cos 𝐴 sin 𝐵
tan 𝐴 + tan 𝐵
tan 𝐴 + 𝐵 =
1 − tan 𝐴 tan 𝐵
cos 𝐴 − 𝐵 = cos 𝐴 cos 𝐵 + sin 𝐴 sin 𝐵
sin 𝐴 − 𝐵 = sin 𝐴 cos 𝐵 − cos 𝐴 sin 𝐵
tan 𝐴 − tan 𝐵
tan 𝐴 − 𝐵 =
1 + tan 𝐴 tan 𝐵
Choose three examples from the chart (one sine, cosine, and tangent) to apply sum and difference identities.
1.
2.
3.
Since the “formulas” are identities, they can be used forward and backwards. Tell what each expression is
equivalent to.
𝜋
𝜋
𝜋
𝜋
cos cos − sin sin
9
7
9
7
sin 60° cos 10° − cos 60° sin 10°
Establish the following identity through graphing, table analysis, and formal algebraic proof.
sin
Graph
𝜋
− 𝑥 = cos 𝑥
2
Table
Algebraic Proof
Establish the following identity through formal proof. Justify each step.
cos 𝑥 + 𝑦 + cos 𝑥 − 𝑦
= 2 cot 𝑥
sin 𝑥 cos 𝑦
Statements
Reasons
Apply the sum identities to each of the following:
sin 𝐴 + 𝐴 =
sin 2𝐴 =
Ex. 1: If
cos 𝐴 + 𝐴 =
tan 𝐴 + 𝐴 =
cos 2𝐴 =
cos 𝜃 =
<
=>
,
>?
@
< 𝜃 < 2𝜋, then find:
tan 2𝐴 =
Ex. 2: Simplify each expression. Evaluate if possible.
A) cos 2𝜃 =
A) cos @ 7𝑥 − sin@ 7𝑥 =
B) 2 cos @ 15° − 1 =
C) 6 sin 𝑥 cos 𝑥 =
B) sin 2𝜃 =
C) tan 2𝜃 =
D) @ BCD EF°
=GBCDH EF°
=
There is one last set of identities we would like to derive. Start with two versions of the double-angle identities
for cosine. Then solve each for either cos 𝐴 or sin 𝐴.
cos 2𝐴 = 2 cos @ 𝐴 − 1
Go back and substitute
cos
𝐴
=
2
J
@
for
cos 2𝐴 = 1 − 2 sin@ 𝐴
𝐴.
sin
Ex. 1: Find the exact value of
𝐴
=
2
tan
sin 22.5°.
@
>?
>
@
Ex. 3: Given cos 𝜃 = , with
< 𝜃 < 2𝜋,
Ex. 2: Simplify
find
Q
cos .
@
𝐴
=
2
=GLMN <O°
NPD <O°
.
UNIT 3B PRACTICE
1. Find the exact value for each trig expression. Use the identities introduced in this unit; for many, there
are multiple ways to evaluate.
R?
B.
A.
cos
D.
cos 22.5°
sin 180° + 𝜃
=@
E.
2. Find the exact value for
A.
tan
R?
=@
sin
R?
T
C.
F.
using three different strategies.
B.
C.
3. Complete the table.
Given
A.
• sin 𝐴 =
U
<
• cos 𝐵 = −
• ?
@
Find
A.
<
=>
<𝐴<𝜋
• 𝜋 < 𝐵 <
• sin 𝐴 + 𝐵
• tan 𝐴 − 𝐵
>?
@
• cos 2𝐴
• cos 𝐴 + 𝐵
BCD @F°SBCD @<°
=GBCD @F° BCD @<°
sin
?
=@
?
?
U
=@
cos + cos
sin
?
U
B.
B.
• sin 𝜃 = −
• cos 𝜃 > 0
<
• cos 2𝜃
R
• tan 2𝜃
C.
Q
• cos
Q
• sin
J
• cos
J
• tan
J
C.
• sec 𝐴 = 2
• • tan
A in Quadrant IV
@
@
@
@
@
• sin 2𝐴
4. Use graphing or check values to determine whether each statement is true or false. Be sure to show
support your conclusion.
A. sin 2𝐴 = 2 sin 𝐴
B. sin 𝑥 −
C. If
?
@
==?
@
= cos 𝑥
Q
< 𝜃 < 𝜋, then cos < 0
@
D.
sin 𝑥 + 𝑦 = sin 𝑥 + sin 𝑦
E.
6 sin 𝑥 cos 𝑥 = 3 sin 2𝑥
F.
4 sin 45° cos 15° = 1 + 3
5. Establish each identity. Justify each step.
A. B. NPD JSX
LMN J LMN X
= tan 𝐴 + tan 𝐵
4 sin 𝑥 cos 𝑥 + 2 = 2 sin 2𝑥 + 2