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Power-Reducing Identities
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Double-Angle and Power-Reducing
IdentitiesSection 5.4
Double-Angle Identitiessin 2 2sin cosu u u
2 2
2
2
cos sin
cos 2 2cos 1
1 2sin
u u
u u
u
2
2 tantan 21 tan
uuu
Proving the first of these:
sin 2u sin u u sin cos cos sinu u u u 2sin cosu u
Power-Reducing Identities2 1 cos 2sin
2uu
2 1 cos 2cos2uu
2 1 cos 2tan1 cos 2
uuu
Guided Practice2cos2 2cos 1u u Prove the given identity.
cos 2u cos u u cos cos sin sinu u u u
2 2cos sinu u
2 2cos 1 cosu u 22cos 1u
Guided Practice2
2 tantan 21 tan
uuu
Prove the given identity.
tan 2u tan u u tan tan1 tan tan
u uu u
2
2 tan1 tan
uu
Guided Practice4 4cos sin cos 2 Prove the given identity.
2 2 2 2cos sin cos sin 4 4cos sin
2 21 cos sin
cos 2
Guided Practice4cos xRewrite in terms of trigonometric functions with
no power greater than 1.
22cos x4cos x21 cos2
2x
21 2cos2 cos 2
4x x
1 1 1 1 cos 4cos 24 2 4 2
xx
1 1 1 1cos 2 cos 44 2 8 8
x x 1 3 4cos 2 cos 48
x x
Guided Practice 0,2Find all solutions to the given equation in the interval .
sin 2 sinx xsin 2 sin 0x x
2sin cos sin 0x x x sin 2cos 1 0x x
sin 0x or1cos2
x
50, , ,3 3
x
Guided Practice 0,2Find all solutions to the given equation in the interval .
cos 2 cosx xcos2 cos 0x x
22cos 1 cos 0x x 22cos cos 1 0x x
1cos2
x or cos 1x
2 40, ,3 3
x
2cos 1 cos 1 0x x
Guided Practice 0,2Find all solutions to the given equation in the interval .
2cos cos cos 2x x x 2 2cos cos 2cos 1x x x
20 cos cos 1x x
1 5cos2
x
Quadratic Formula: Only keep this answer:
1 5cos2
x
Guided Practice 0,2Find all solutions to the given equation in the interval .
2cos cos cos 2x x x
or
1 5cos2
x
1 1 5cos 2.2372
x
1 1 52 cos 4.046
2x
Half-Angle Identities
Half-Angle Identities
1 cossin2 2u u
1 coscos2 2u u
1 cos1 cos
1 costan2 sin
sin1 cos
uu
u uuuu
Guided Practice
tan195
Use half-angle identities to find an exact value of the givenexpression.
1 cos390sin390
1 cos30sin 30
1 3 21 2
2 3
Guided Practice
5sin12
Use half-angle identities to find an exact value of the givenexpression.
1 cos 5 62
cos 5 612 2
1 32 4
1 2 34
1 2 32
Since , we take the positive value…5 1sin 2 312 2
5sin 012
Guided Practice
cos8
Use half-angle identities to find an exact value of the givenexpression.
1 cos 42
1 21
2 2
1 2 24
1 2 22
Since , we take the positive value…1cos 2 2
8 2
cos 08
Guided Practice
sin 2 cos
Write the given expression as one involving only and .
(Note: There are multiple correct answers!!!)
2sin cos cos
cos 2sin 1
sin cos
Guided Practice
sin 2 cos3 sin 2 cos 2 2sin cos cos 2 cos sin 2 sin
2 2 22sin cos cos sin cos 2sin cos 3 22sin cos cos 3sin cos
Write the given expression as one involving only and .
(Note: There are multiple correct answers!!!)
sin cos
Guided Practice2cos6 2cos 3 1x x Prove the identity:
cos6x cos 2 3x22cos 3 1x
Guided Practice2cot 2 cot tanx x x Prove the identity:
2cot 2x2
tan 2x
2
22 tan1 tan
xx
22 1 tan
2 tan
x
x
1 tantan
xx
cot tanx x
Guided Practice 2sin 3 sin 3 4sinx x x Prove the identity:
sin 3x sin 2x x sin 2 cos cos 2 sinx x x x
2 22sin cos 1 2sin sinx x x x
2 2sin 2cos 1 2sinx x x
2 2sin 2 2sin 1 2sinx x x
2sin 3 4sinx x
Guided Practice 2sin 4 4sin cos 2cos 1x x x x Prove the identity:
sin 4x sin 2 2x
2sin 2 cos 2x x
22 2sin cos 2cos 1x x x
24sin cos 2cos 1x x x