Download 14-6 - Ithaca Public Schools

Name_____________________________________ Class____________________________ Date________________
Lesson 14-6
Angle Identities
Lesson Objectives
1 Verifying and using angle identities
2 Verifying and using sum and
difference identities
NAEP 2005 Strand: Algebra
Topic: Variables, Expressions, and Operations
Local Standards: ____________________________________
Key Concepts.
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Angle Identities
Negative Angle Identities
sin (θ) cos (θ) tan (θ) Cofunction Identities
sin (p
2 θ) cos (p
2 θ) tan (p
2 θ) Angle Difference Identities
sin (A B) sin
cos
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cos (A B) cos
cos
sin
sin
sin
cos
sin
sin
sin
cos
tan (A B) tan
1 tan
tan
tan
Angle Sum Identities
sin (A B) sin
cos
cos (A B) cos
cos
tan (A B) tan
1 tan
tan
tan
Examples.
(
(
1 Verifying Angle Identities Use the fact that π θ p2 θ p2
the identity sin (π θ ) sin θ.
(
(
p
sin (π θ ) sin p
2 θ2
cos
(
Daily Notetaking Guide
))
(θ p2 )
)
θ
)) to verify
Take the sine of each side.
Cofunction identity
Negative angle identity
Cofunction identity
Algebra 2 Lesson 14-6
289
Name_____________________________________ Class____________________________ Date ________________
(
)
(
( π2
θ .
)
2 Solving Trigonometric Equations Solve cos p2 θ sin p2 θ for 0 θ 2π.
(π )
1
π
sin ( 2 θ )
cos 2 θ
θ
1
)
Cofunction identities
θ 1
Tangent identity
θ Solve for θ.
π
Another solution is
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θ
Divide by sin
.
3 Cofunction Identities in a Right Triangle Find a cofunction identity for
tan (90° A), where A is an acute angle of a right triangle.
In a right triangle, the
their measures is
angles are complementary (the sum of
). So A B and B A, where A
and B are the measures of the acute angles.
(90° A)
(90° A)
A
Tangent identity
Cofunction identities
A
A
Cotangent identity
Quick Check.
1. Verify the identity
(
)
cos θ p
2 sin θ.
290
Algebra 2 Lesson 14-6
(
)
2. Solve sin p
2 θ sec θ
for 0 θ 2π.
3. Derive a cofunction identity
for sec (90° A).
Daily Notetaking Guide
© Pearson Education, Inc., publishing as Pearson Prentice Hall.
tan (90° A) Name_____________________________________ Class____________________________ Date ________________
4 Using Angle Difference Identities Find the exact value of cos 165°.
You know exact values for 180°, 60°, and 45°, and that 165° 180° (60° 45°).
cos (A B) B
A
A
B
cos (180° (60° 45°))
(
) cos (
) cos (
(
)(cos
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(
(
) sin
°sin (
) ( ) sin (
)
)
° cos (
cos
° sin
° cos
° sin
for B.
° for A
Substitute
°)
° for B.
and
Replace with exact values.
Simplify.
in Quadrant II, cos 165° Since the cosine is
© Pearson Education, Inc., publishing as Pearson Prentice Hall.
and
Simplify.
° for A
Use the exact values for
cos 180° and sin 180°.
Substitute
)
( )( ) ( )( )
(
) (
)
)
Cosine Angle Difference Identity
.
5 Using Angle Sum Identities Find the exact value of sin 195°.
Use the fact that 195° 135° 60°.
sin (A B) B
A
cos
cos
sin (135° 60°) sin
( ) (
A
)( )
B
sin
Sine Angle Sum Identity
°
Substitute
for A and
° for B.
Replace with
exact values.
Simplify.
Quick Check.
4. Find the exact value of sin 15°.
Daily Notetaking Guide
5. Find the exact value of tan 105°.
Algebra 2 Lesson 14-6
291