Download Lesson 14-7 - Ithaca Public Schools

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Lesson 14-7
Double-Angle and Half-Angle Identities
Lesson Objectives
1 Verifying and using double angle
identities
2 Verifying and using half-angle
identities
NAEP 2005 Strand: Algebra
Topic: Variables, Expressions, and Operations
Local Standards: ____________________________________
Key Concepts.
2
cos 2θ 2
cos 2θ 2
cos 2θ 2
θ θ 2
2
θ
θ
tan 2θ θ
θ
sin 2θ 2
2
θ
1
2
θ
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Double-Angle Identities
1
sin A 2
A
2
cos A 2
1
tan A 2
1
A
1
A
A
2
Examples.
1 Using a Double-Angle Identity Use a double-angle identity to find the exact
value of sin 600°.
sin 600° sin (2 300°)
2
(
2
292
Rewrite 600 as (2 300).
300°
)(
Algebra 2 Lesson 14-7
300°
)
Use a sine
-angle identity.
Replace with exact values. Simplify.
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Half-Angle Identities
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2 Verifying an Identity Verify the identity
sin 2θ
(1 sin2 θ)
θ 2
θ
-angle and
θ
identities
θ
2
Simplify.
θ
2
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sin 2u
2 tan θ.
(1 2 sin 2 u)
Identity
θ
3 Using Half-Angle Identities Use the half-angle identities to find the exact
value of sin 75˚.
sin 75 sin 150
Rewrite 75 as
2
(
150
2
1 (
2
4
)
)
( ).
150
2
Use the
since sin 75 is positive.
Substitute the exact value for
square root,
150.
Simplify.
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Simplify.
Quick Check.
1. Use a double-angle identity to find the exact value of sin 120°.
2. Verify the identity 2 cos 2θ 4 cos2 θ 2.
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Algebra 2 Lesson 14-7
293
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Example.
12 and 90° θ 180°, find sin u.
4 Using Half-Angle Identities Given cos θ 13
2
Since 90° θ 180°,
2u θ
(
2
-angle identity
2
1 25
26
26
26
.
)
Substitute. Choose the
square root since
θ
2
is in Quadrant
.
Simplify.
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sin θ 2
and 2u is in Quadrant
Rationalize the denominator.
Quick Check.
4. Use the information from Example 4 to find exact values of cos 2u and tan 2u.
294
Algebra 2 Lesson 14-7
Daily Notetaking Guide
© Pearson Education, Inc., publishing as Pearson Prentice Hall.
3. Use the half-angle identities to find the exact value of each expression.
a. sin 150°
b. tan 150°