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Regents Exam Questions A2.A.77: Double Angle Identities 3
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A2.A.77: Double Angle Identities 3: Apply the double-angle and half-angle formulas for
trigonometric functions
1 If sinA 
1)

2)
2
3
3)

4)
7
9
2)
3)
4)
4 If sinA 
2
3
1)
2)
7
9
2 If cos  
1)
1
, what is the value of cos 2A?
3
3)
4)
3
, then what is cos 2 ?
4
9
64
1
4
23
32
55
64
5 If x is an acute angle and sin x 
12
, then cos 2x
13
equals
25
1)
169
119
2)
169
25
3) 
169
119
4) 
169
1
8
9
16
1

8
3
2
3 If x is an acute angle, and cos x 

3
, what is the value of cos 2A?
8
4
, then cos 2x is
5
6 If  is an acute angle such that sin  
equal to
6
1)
25
1
2)
25
2
3)
25
7
4)
25
the value of sin 2 ?
12
1)
13
10
2)
26
60
3)
169
120
4)
169
1
5
, what is
13
Regents Exam Questions A2.A.77: Double Angle Identities 3
Name: _______________________
www.jmap.org
7 If x is a positive acute angle and sinx 
1
, what is
2
5
and A is in Quadrant I. Find, in
3
simplest form, the value of sin2A and cos 2A.
12 SinA 
sin2x?
1
1) 
2
1
2)
2
3)

13 The expression 1  2 sin 2 30 has the same value as
1) sin60
2) cos 60
3) cos 15
4) sin15
3
2
3
2
4)
14 The expression 1  2 sin 2 45 has the same value as
1) cos 90
2) cos 45
3) sin90
1
4) sin 22 
2
2
8 If sin A  where 0  A  90, what is the value
3
of sin2A?
2 5
1)
3
2)
2 5
9
3)
4 5
9
4)

15 The expression 2 sin30 cos 30 has the same value
as
1) sin15
2) cos 60
3) sin60
4) cos 15
4 5
9
9 If sin A 
10 If sinA 
3
, find cos 2A.
5
16 The expression cos 2 40  sin 2 40 has the same value
as
1) sin20
2) sin80
3) cos 80
4) cos 20
2
, find cos 2A.
3
17 If  is an obtuse angle and sin   b, then it can be
concluded that
1) tan   b
2) cos   b
3) cos 2  b
4) sin 2  b
3
11 If  is in Quadrant II and cos    , find an exact
4
value for sin 2 .
2
ID: A
A2.A.77: Double Angle Identities 3: Apply the double-angle and half-angle formulas for
trigonometric functions
Answer Section
1 ANS: 4
2
ÁÊ 1 ˜ˆ
2 7
cos 2A  1  2 sin 2 A  1  2 ÁÁÁÁ ˜˜˜˜  1  
9 9
Ë 3¯
REF: 011311a2
2 ANS: 1
ÊÁ 3 ˆ˜ 2
ÊÁ 9 ˆ˜
9 8 1
cos 2  2 ÁÁÁÁ ˜˜˜˜  1  2 ÁÁÁÁ ˜˜˜˜  1   
8 8 8
Ë4¯
Ë 16 ¯
REF: 081522a2
3 ANS: 4
REF: fall9905b
4 ANS: 3
2
ÁÊ 3 ˜ˆ
32 9
23
cos 2A  1  2sin 2 A  1  2 ÁÁÁÁ ˜˜˜˜ 


8
32
32
32
Ë ¯
REF: 011510a2
5 ANS: 4
REF: 010418b
6 ANS: 4
If  is an acute angle and sin  
5
12
, cos   .
13
13
.
REF: 060413b
7 ANS: 4
If x is an acute angle and sin x 
3
1
, cos x 
.
2
2
.
REF: 060604b
1
ID: A
8 ANS: 3
2
ÁÊÁ 2 ˜ˆ˜
ÁÁ ˜˜  cos 2 A  1
Á3˜
Ë ¯
sin2A  2 sin A cos A
ˆ
Ê
ÁÊÁ 2 ˜ˆ˜ ÁÁÁÁ 5 ˜˜˜˜
˜˜
 2 ÁÁÁ ˜˜˜ ÁÁ
Ë 3 ¯ ÁÁË 3 ˜˜¯
5
cos 2 A 
9
cos A  
5
, sin A is acute.
3

4 5
9
REF: 011107a2
9 ANS:
7
25
REF: 068817siii
10 ANS:
1
9
REF: 088713siii
11 ANS:
3 7
8
REF: 089940siii
12 ANS:
4 5
1
,
9
9
13
14
15
16
17
REF:
ANS:
ANS:
ANS:
ANS:
ANS:
If
088938siii
2
1
3
3
4
REF:
REF:
REF:
REF:
068523siii
089521siii
069727siii
089821siii
is an obtuse angle, cos
.
is negative, sin
is negative.
is positive, and b is positive.
REF: 060118b
2
is negative.