Lecture 16 section 6.2 sum-difference identities
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Transcript of Lecture 16 section 6.2 sum-difference identities
MATH 107
Section 6.2
Sum and Difference Formulas
2© 2011 Pearson Education, Inc. All rights reserved
SUM AND DIFFERENCE FORMULASFOR COSINE
cos cos cos sin sin
cos cos cos sin sin
u v u v u v
u v u v u v
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EXAMPLE 1 Using the Difference Identity for Cosine
Find the exact value of by using
Solution
cos cos12 3 4
12cos
Example
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Find the exact value of cos15 .
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SUM AND DIFFERENCE IDENTITIESFOR SINE
vuvuvu
vuvuvu
sincoscossinsin
sincoscossinsin
19Find the exact value of sin .
12
(Answer on next slide.)
19Find the exact value of sin .
12
19 16 3 4sin sin sin
12 12 12 3 4
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EXAMPLE 5 Using the Sum Formula for Sine
Find the exact value of
without using a calculator.
sin 63ºcos27º cos63ºsin 27º sin 63º 27º
sin 90º
Solution
This expression is the right side of the sum identity for sin (u + v), where u = 63º and v = 27º.
Example
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10© 2011 Pearson Education, Inc. All rights reserved
.sin2
cos vv
BASIC COFUNCTION IDENTITIES
If v is any real number or angle measured in radians, then
If angle v is measured in degrees, then replace by 90º in these identities.
.cos2
sin vv
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EXAMPLE 3 Using Cofunction Identities
Prove that for any real number x, tan cot .2
x x
Solution
cos
sin
x
x
cot x
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SUM AND DIFFERENCE IDENTITIESFOR TANGENT
tan tantan
1 tan tan
tan tantan
1 tan tan
u vu v
u v
u vu v
u v
13© 2011 Pearson Education, Inc. All rights reserved
EXAMPLE 11 Verifying an Identity
Verify the identity .
Solution
Apply the difference identity.
tan tantan
1 tan tan
xx
x
0 tantan
1 0 tan
xx
x
tan tanx x
Therefore, the given equation is an identity.