Regents Exam Questions
A2.A.58: Reciprocal Trigonometric Relationships
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A2.A.58: Reciprocal Trigonometric Relationships: Know and apply the co-function and
reciprocal relationships between trigonometric ratios
1 If csc   2, what is the value of sin  ?
1) 2
2) 2
1
3) 
2
1
4)
2
5 The expression sec 2 x  csc 2 x is equivalent to
1) 1
1
2)
cos x sin x
3) cos 2 x sin 2 x
1
4)
cos 2 x sin 2 x
1
, a  0, which statement must be true?
a
csc x  a
1
csc x  
a
sec x  a
1
sec x  
a
2 If sinx 
1)
2)
3)
4)
6 The expression sec 2   tan 2  is equal to
1) 1
2) 0
3) sin 2 
1
4)
cos 2 
7 The expression cot   sec  is equivalent to
cos 
1)
sin 2 
sin 
2)
cos 2 
3) csc 
4) sin 
3 The expression 1  sec x is equivalent to
1) tan x
cos x  1
2)
cos x
sin x  1
3)
sin x
tan x
4)
sec x  1
8 The expression (tan )(csc ) is equivalent to
1) cos 
2) sec 
3) csc 
4) csc  cot 
4 For all values of x for which the expressions are
defined, sec x  tan x is equivalent to
1) 1
2) cos x  cot x
1  sinx
3)
cos x
4)
9 Expressed in simplest form, csc   tan   cos  is
equivalent to
1) 1
2) sin 
3) cos 
4) tan 
cos x  sin 2 x
sin x cos x
1
Regents Exam Questions
A2.A.58: Reciprocal Trigonometric Relationships
Name: ________________________
www.jmap.org
10 The expression (sec 2 )(cot 2 )(sin ) is equivalent
to
1) sin 
2) cos 
3) csc 
4) sec 
16 The expression
1)
2)
3)
4)
11 The expression cos y(csc y  sec y) is equivalent to
1) cot y  1
2) tan y  1
3) 1  tan y
4) cos y
3)
4)
12 The expression sin (cot   csc ) is equivalent to
1)
2)
3)
4)
cos   sin 2 
2 cos 
sin 
cos   1
13 Express cos (sec   cos ), in terms of sin  .
2)
3)
4)
14 Which trigonometric expression does not simplify
to 1?
1) sin2 x(1  cot 2 x)
3)
4)
sin 
cos 
sin 
cos 
cos 
sin 
18 The expression
1)
2)
sin x
cos x
tan x
sec x
17 The expression
1)
2)
cot x
is equivalent to
csc x
sec 
is equivalent to
csc 
tan 
is equivalent to
sec 
cos 2 
sin 
sin 
cos 2 
cos 
sin 
19 For all values of  for which the expression is
csc 
is equivalent to
defined,
sec 
1) cos 
2) sin 
3) cot 
4) tan 
sec 2 x(1  sin 2 x)
cos 2 x(tan 2 x  1)
cot 2 x(sec 2 x  1)
15 The expression (1  cos x)(1  cos x) is equivalent
to
1) 1
2) sec 2 x
3) sin 2 x
4) csc 2 x
cot x sin x
as a single trigonometric
sec x
function, in simplest form, for all values of x for
which it is defined.
20 Express
2
Regents Exam Questions
A2.A.58: Reciprocal Trigonometric Relationships
Name: ________________________
www.jmap.org
ÊÁ
3 ˆ˜
26 What is the value of csc ÁÁÁÁ Arc sin ˜˜˜˜ ?
4¯
Ë
3
1)
4
4
2)
3
sin 2 x  cos 2 x
is equal to
21 The expression
cos x
1) csc x
2) sec x
3) cos x  tan x
4) sinx  cos x  tan x
3)
22 The expression
1)
2)
3)
4)
sin 2   cos 2 
is equivalent to
1  sin 2 
cos 2 
sin 2 
sec 2 
csc 2 
23 Show that
ˆ
ÊÁ
ÁÁ
3 ˜˜˜˜
Á
˜
27 Evaluate: csc ÁÁ Arc sin
ÁÁ
2 ˜˜˜
¯
Ë
sec 2 x  1
is equivalent to sin 2 x.
sec 2 x
24 The expression sin A 
1)
2)
3)
4)
2)
3)
4)
ÊÁ
5 ˆ˜
28 What is the value of sec ÁÁÁ Arc cos ˜˜˜˜ ?
ÁË
7¯
cos 2 A
is equivalent to
sinA
1
sin A
sec A
csc A
25 The expression
1)
4)
7
4
4
7
sin 2 B
 cos B is equivalent to
cos B
1
1
cos B
1
sec B
sin 2 B
3
ID: A
A2.A.58: Reciprocal Trigonometric Relationships: Know and apply the co-function and
reciprocal relationships between trigonometric ratios
Answer Section
1 ANS: 3
REF: 080703b
2 ANS: 1
1
sin x 
.
csc x
REF: 060904b
3 ANS: 2
4
5
6
7
REF:
ANS:
ANS:
ANS:
ANS:
080813b
3
4
1
3
8
9
10
11
12
13
REF:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
010915b
2
1
3
1
4
cos  
1
 cos 2   1  cos 2   sin 2 
cos 
REF: 068623siii
REF: 089428siii
REF: 060220siii
REF:
REF:
REF:
REF:
REF:
010122siii
069921siii
010427siii
068731siii
060018siii
REF: 061230a2
14 ANS: 3
ˆ
ÁÊÁ
ÁÊ sin 2 x
˜ˆ˜
cos 2 x ˜˜˜˜
1
2
2
2
2 Á
ÁÁ
˜˜  sin 2 x  cos 2 x  1
sin 2 x ÁÁÁÁ 1 
x

cos
x

1
cos
x

1

sin
(cos
x)

1
˜
Á
˜˜
2
2
˜˜
ÁÁ cos 2 x
Á
˜
sin
cos
x
x
Ë
¯
Ë
¯
2
2
˜ˆ
1
cos x
cos x ÁÊÁÁ 1
 1 ˜˜˜˜ 

 csc 2 x  cot x  1
ÁÁ
2
2
2
2
sin
sin x Ë cos x
sin
x
x
¯
REF: 011515a2
1
ID: A
15 ANS: 3
REF: 010608b
16 ANS: 2
cos x
sinx
cot x

 cos x
1
csc x
sinx
REF: 061410a2
17 ANS: 3
REF: 010402b
18 ANS: 4
REF: 010508b
19 ANS: 3
REF: 080318siii
20 ANS:
cos x
sin x
sinx
cot x sin x

 cos 2 x
1
sec x
cos x
REF: 061334a2
21 ANS: 2
REF: 089316siii
22 ANS: 3
sin 2   cos 2 
1

 sec 2 
1  sin 2 
cos 2 
REF: 061123a2
2
ID: A
23 ANS:
1
1
cos 2 x
cos 2 x 1  cos 2 x


 sin 2 x
2
1
1
cos x
cos 2 x
REF: 081533a2
24 ANS: 4
REF: 060720b
25 ANS: 2
26 ANS: 2
REF: 019530siii
REF: 080817b
27 ANS:
2
3
REF: 069410siii
28 ANS:
7
5
REF: 010315siii
3