Download Classwork and Key - Garnet Valley School District

Algebra II Trigonometry Honors
Unit 10: Day 9 Notes: Double Angle Identities
Classwork
Find the exact value of each.
3
where 270 ≤ θ < 360
5
Find sin 2θ
2) cos θ =
3
where 630 ≤ θ < 720
5
Find tan 2θ
4
5π
where 2π ≤ θ <
3
2
Find cos 2θ
4) sin θ =
1) cos θ =
12
π
where 0 ≤ θ <
13
2
Find tan 2θ
3) tan θ =
Verify each identity.
5) 2tan 2 xsin xcos x =
sin 2x
6)
2
cot x
2sin xcos x
= tan 2x
cos 2x
7) cos 2 x(1 − cos 2x) = sin 2 x(1 + cos 2x)
8) sec 2 x − 2sin 2 x = cos 2x + tan 2 x
9) tan 2 xcos 2 x = cos 2 x − cos 2x
tan 2 x
sin 2 x
10)
=
1 − tan 2 x cos 2x
11)
tan 2 x
1
=
1 − cos 2x 2cos 2 x
12) sin 2x + tan xcos x = sin x ⋅ (2cos x + 1)
13) 2cot x ⋅ (1 − cos 2x) = 2sin xcos x + sin 2x
14)
15)
sin 2xsec 2 x
2
tan x
= 2csc 2 xsin xcos x
1 − tan 2 x sec 2 x
=
sin 2x
tan 2x
16) 2(sin xcos xcsc 2 x − 1) =
sin 2x − 1 + cos 2x
sin 2 x
Worksheet by Kuta Software LLC
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Answers to Classwork
1) −
24
25
2)
2
5) 2tan xsin xcos x
2sin xcos x
cot 2 x
sin 2x
cot 2 x
24
7
Use cot x =
7
25
2sin xcos x
6)
cos 2x
3) −
1
tan x
4) −
Use sin 2x = 2sin xcos x
Use sin 2x = 2sin xcos x
sin 2x
cos 2x
Use tan 2x =
■
tan 2x
■
2
7) cos x(1 − cos 2x)
(1 − cos 2x)(1 + cos 2x)
2
sin 2 x(1 + cos 2x)
2
2
8) sec x − 2sin x
Use cos 2 x =
1 + cos 2x
2
Use sin 2 x =
1 − cos 2x
2
120
119
sin 2x
cos 2x
■
Use tan 2 x + 1 = sec 2 x
tan 2 x + 1 − 2sin 2 x
Use cos 2x = 1 − 2sin 2 x
cos 2x + tan 2 x
■
2
2
9) tan xcos x
( )
sin x
cos x
Decompose into sine and cosine
2
⋅ cos 2 x
Simplify
sin 2 x
Use cos 2x = cos 2 x − sin 2 x
cos 2 x − cos 2x
■
2
10)
tan x
Decompose into sine and cosine
1 − tan 2 x
( )
( )
sin x
cos x
2
sin x
cos x
1−
2
sin 2 x
2
2
cos x − sin x
sin 2 x
cos 2x
Simplify
Use cos 2x = cos 2 x − sin 2 x
■
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11)
tan 2 x
1 − cos 2x
Use tan 2 x =
1 − cos 2x
(1 − cos 2x)(1 + cos 2x)
Cancel common factors
1
1 + cos 2x
Use cos 2x = 2cos 2 x − 1
1
■
2cos 2 x
12) sin 2x + tan xcos x
Use sin 2x = 2sin xcos x
cos x ⋅ (2sin x + tan x)
(
cos x ⋅ 2sin x +
sin x
cos x
Decompose into sine and cosine
)
sin x ⋅ (2cos x + 1)
13) 2cot x ⋅ (1 − cos 2x)
14)
1 − cos 2x
1 + cos 2x
Simplify
■
Use cos 2x = 1 − 2sin 2 x
cos x
sin x
4cot xsin 2 x
Use cot x =
4cos xsin 2 x
sin x
Cancel common factors
4sin xcos x
Use sin 2x = 2sin xcos x
2sin xcos x + sin 2x
sin 2xsec 2 x
■
tan 2 x
( )
( )
1
sin 2x ⋅
cos x
sin x
cos x
Decompose into sine and cosine
2
2
sin 2x
sin 2 x
2sin xcos x
2
sin x
2csc 2 xsin xcos x
Simplify
Use sin 2x = 2sin xcos x
Use csc x =
1
sin x
■
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15)
1 − tan 2 x
sin 2x
( )
sin x
1−
cos x
sin 2x
Decompose into sine and cosine
2
Simplify
cos 2 x − sin 2 x
cos 2 xsin 2x
cos 2x
Use cos 2x = cos 2 x − sin 2 x
Use sec x =
2
cos xsin 2x
sec 2 xcos 2x
sin 2x
Use tan 2x =
sec 2 x
■
tan 2x
2
16) 2(sin xcos xcsc x − 1)
(
( ) )
1
2 sin xcos x ⋅
sin x
2sin x ⋅ (cos x − sin x)
sin 2 x
sin 2x − 2sin 2 x
2
sin x
sin 2x − 1 + cos 2x
sin 2x
cos 2x
Decompose into sine and cosine
2
2(−sin x + cos x)
sin x
sin 2 x
1
cos x
−1
Simplify
Create a common factor
Use sin 2x = 2sin xcos x
Use cos 2x = 1 − 2sin 2 x
■
Worksheet by Kuta Software LLC
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