ECAT Mathematics
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ECAT Mathematics
Chapter 10: Trigonometric Identities
84 Questions
90 Minutes
Pass: 60%+
ECAT Prep
0
sin(A+B) =
A
sinA cosB + cosA sinB
B
sinA cosB − cosA sinB
C
sinA sinB + cosA cosB
D
cosA cosB − sinA sinB
1
sin(A−B) =
A
sinA cosB + cosA sinB
B
sinA cosB − cosA sinB
C
sinA sinB − cosA cosB
D
cosA cosB + sinA sinB
2
cos(A+B) =
A
cosA cosB − sinA sinB
B
cosA cosB + sinA sinB
C
sinA cosB + cosA sinB
D
sinA cosB − cosA sinB
3
cos(A−B) =
A
cosA cosB − sinA sinB
B
cosA cosB + sinA sinB
C
sinA cosB − cosA sinB
D
sinA cosB + cosA sinB
4
tan(A+B) =
A
(tanA+tanB)/(1+tanAtanB)
B
(tanA+tanB)/(1−tanAtanB)
C
(tanA−tanB)/(1+tanAtanB)
D
tanA tanB
5
tan(A−B) =
A
(tanA+tanB)/(1−tanAtanB)
B
(tanA−tanB)/(1+tanAtanB)
C
(tanA−tanB)/(1−tanAtanB)
D
tanA−tanB
6
sin 2A =
A
sin²A−cos²A
B
2sinAcosA
C
cos²A−sin²A
D
1−2sin²A
7
cos 2A =
A
2sinAcosA
B
sin²A−cos²A
C
cos²A−sin²A
D
2cos²A
8
cos 2A can also be written as:
A
1−2sin²A
B
2cos²A−1
C
cos²A−sin²A
D
All of the above
9
tan 2A =
A
2tanA/(1+tan²A)
B
2tanA/(1−tan²A)
C
tanA/(1−tan²A)
D
2tanA
10
sin²A = (1−cos2A)/?
A
2
B
4
C
3
D
1
11
cos²A = (1+cos2A)/?
A
2
B
4
C
3
D
1
12
sin(A+B)sin(A−B) =
A
sin²A−sin²B
B
cos²B−cos²A
C
sin²A+sin²B
D
cos²A−cos²B
13
2sinAcosB =
A
sin(A+B)+sin(A−B)
B
sin(A+B)−sin(A−B)
C
cos(A−B)−cos(A+B)
D
cos(A+B)+cos(A−B)
14
2cosAsinB =
A
sin(A+B)+sin(A−B)
B
sin(A+B)−sin(A−B)
C
cos(A−B)−cos(A+B)
D
cos(A+B)+cos(A−B)
15
2cosAcosB =
A
cos(A+B)+cos(A−B)
B
cos(A−B)−cos(A+B)
C
cos(A+B)−cos(A−B)
D
sin(A+B)+sin(A−B)
16
2sinAsinB =
A
cos(A−B)−cos(A+B)
B
cos(A+B)−cos(A−B)
C
sin(A+B)+sin(A−B)
D
sin(A−B)−sin(A+B)
17
sinC+sinD = 2sin((C+D)/2)?
A
cos((C−D)/2)
B
sin((C−D)/2)
C
cos((C+D)/2)
D
sin((C+D)/2)
18
sinC−sinD = 2cos((C+D)/2)?
A
cos((C−D)/2)
B
sin((C−D)/2)
C
cos((C+D)/2)
D
tan((C−D)/2)
19
cosC+cosD = 2cos((C+D)/2)?
A
cos((C−D)/2)
B
sin((C−D)/2)
C
sin((C+D)/2)
D
tan((C−D)/2)
20
sin3A =
A
3sinA−4sin³A
B
4sinA−3sin³A
C
3sinA+4sin³A
D
sinA(3−4sin²A)
21
cos3A =
A
4cos³A−3cosA
B
3cosA−4cos³A
C
4cos³A+3cosA
D
cosA(4cos²A−3)
22
tan3A =
A
(3tanA−tan³A)/(1−3tan²A)
B
3tanA/(1−tan²A)
C
tanA(3−tan²A)
D
(3tanA+tan³A)/(1+3tan²A)
23
sin A = 2sin(A/2)cos(A/2). This is:
A
Double angle formula
B
Half angle formula written in terms of A/2
C
Product formula
D
Sum formula
24
Half-angle: sin(A/2) =
A
±√((1−cosA)/2)
B
±√((1+cosA)/2)
C
(1−cosA)/2
D
sinA/2
25
Half-angle: cos(A/2) =
A
±√((1−cosA)/2)
B
±√((1+cosA)/2)
C
cosA/2
D
(1+cosA)/2
26
tan(A/2) =
A
sinA/(1+cosA)
B
sinA/(1−cosA)
C
(1−cosA)/sinA
D
Both a and c are equal
27
sin 75° = sin(45°+30°) =
A
(√6+√2)/4
B
(√6−√2)/4
C
(√3+1)/2√2
D
Both A and C
28
cos 75° =
A
(√6+√2)/4
B
(√6−√2)/4
C
(√3−1)/2√2
D
Both B and C
29
sin 15° =
A
(√6−√2)/4
B
(√6+√2)/4
C
(√3−1)/2√2
D
Both A and C
30
The identity sin²A+cos²A=1 is:
A
Product identity
B
Pythagorean identity
C
Sum identity
D
Reciprocal identity
31
sin(A+B)+sin(A−B) =
A
2sinAcosB
B
2cosAsinB
C
2sinAsinB
D
2cosAcosB
32
cos(A−B)−cos(A+B) =
A
2cosAcosB
B
2sinAsinB
C
2sinAcosB
D
2cosAsinB
33
tan 75° = tan(45°+30°) =
A
2+√3
B
2−√3
C
√3
D
(√3+1)/(√3−1)
34
If sinA = 3/5, cos A = 4/5 (Q1): sin2A =
A
24/25
B
7/25
C
12/25
D
2×3/5×4/5=24/25
35
cos2A for sinA=3/5:
A
7/25
B
1/25
C
−7/25
D
24/25
36
The value of sin 105°:
A
(√6−√2)/4
B
(√6+√2)/4
C
(√2+√6)/4
D
Both B and C
37
Verify: 2sin45°cos45° = sin90°:
A
Yes: 2(√2/2)(√2/2)=1=sin90°
B
No
C
Only approximately
D
Undefined
38
cos 2A = 1−2sin²A implies sin²A =
A
(1+cos2A)/2
B
(1−cos2A)/2
C
cos2A
D
2cos2A
39
Which is NOT a standard identity?
A
sin²+cos²=1
B
1+tan²=sec²
C
sinA+cosA=1
D
1+cot²=csc²
40
sin(90°+θ) =
A
sinθ
B
cosθ
C
−sinθ
D
−cosθ
41
cos(90°+θ) =
A
cosθ
B
sinθ
C
−cosθ
D
−sinθ
42
4sinAcosAcos2A =
A
sin4A
B
cos4A
C
sin2A
D
2sin2A
43
sin A cos B = ?
A
(sin(A+B)+sin(A−B))/2
B
sin(A+B)+sin(A−B)
C
(cos(A−B)−cos(A+B))/2
D
cos(A+B)−cos(A−B)
44
tan A + cot A =
A
2/sin2A
B
2sin2A
C
1/sinAcosA
D
Both A and C
45
sin³A = ?
A
(3sinA−sin3A)/4
B
(3sinA+sin3A)/4
C
sin3A/4
D
sinA(1−cos²A)
46
cos³A = ?
A
(3cosA+cos3A)/4
B
(3cosA−cos3A)/4
C
cos3A/3
D
cosA(cos²A−1)
47
The value of sin10°+sin50°−sin70°:
A
1/2
B
0
C
1
D
√3/2
48
cos 36° − cos 72° =
A
1/2
B
1
C
√5/4
D
√3/2
49
If tanA = t, sin2A =
A
2t/(1+t²)
B
(1−t²)/(1+t²)
C
t/(1+t²)
D
2t/(1−t²)
50
If tanA = t, cos2A =
A
2t/(1+t²)
B
(1−t²)/(1+t²)
C
(t²−1)/(t²+1)
D
2t/(1−t²)
51
If tanA = t, tan2A =
A
2t/(1+t²)
B
(1−t²)/(1+t²)
C
2t/(1−t²)
D
t/(1−t²)
52
sinA+sinB = 2sin((A+B)/2)cos((A−B)/2) is:
A
Product formula
B
Sum-to-product formula
C
Difference formula
D
Double angle
53
cosA−cosB = ?
A
2sin((A+B)/2)sin((A−B)/2)
B
−2sin((A+B)/2)sin((A−B)/2)
C
2cos((A+B)/2)sin((A−B)/2)
D
2cos((A+B)/2)cos((A−B)/2)
54
sin²A − sin²B =
A
sin(A+B)sin(A−B)
B
cos(A+B)cos(A−B)
C
cos(A−B)−cos(A+B)
D
sin(A+B)cos(A−B)
55
cos²A − cos²B =
A
−sin(A+B)sin(A−B)
B
cos(A+B)cos(A−B)
C
sin(A+B)sin(A−B)
D
cos²B−cos²A
56
The identity sin(π−x) =
A
sinx
B
−sinx
C
cosx
D
−cosx
57
cos(π−x) =
A
cosx
B
−cosx
C
sinx
D
−sinx
58
Which is the half-angle for tan?
A
±√((1−cosx)/(1+cosx))
B
±√((1+cosx)/(1−cosx))
C
(1+cosx)/sinx
D
Both A and C are equivalent
59
sin(A+B)×sin(A−B) =
A
sin²A−sin²B
B
cos²B−cos²A
C
Both equivalent
D
sin²A×cos²B
60
The product-to-sum formula cosAcosB =
A
(cos(A+B)+cos(A−B))/2
B
(cos(A+B)−cos(A−B))/2
C
cos(A+B)+cos(A−B)
D
sin(A+B)sin(A−B)
61
sinAsinB =
A
(cos(A−B)−cos(A+B))/2
B
(cos(A+B)−cos(A−B))/2
C
cos(A−B)+cos(A+B)
D
cos(A+B)−cos(A−B)
62
sin(α+β)+sin(α−β) =
A
2sinαcosβ
B
2cosαsinβ
C
2sinαsinβ
D
sin2α
63
1−cos2A =
A
2sin²A
B
2cos²A
C
sin2A
D
cos²A
64
1+cos2A =
A
2sin²A
B
2cos²A
C
sin2A
D
1
65
sin2A/(1+cos2A) =
A
tanA
B
cotA
C
secA
D
2tanA
66
(1−cosA)/sinA =
A
tanA
B
cotA
C
tan(A/2)
D
cot(A/2)
67
The identity tan(45°+A) =
A
(1+tanA)/(1−tanA)
B
(1−tanA)/(1+tanA)
C
tanA+1
D
1−tanA
68
The identity tan(45°−A) =
A
(1+tanA)/(1−tanA)
B
(1−tanA)/(1+tanA)
C
tanA−1
D
1+tanA
69
sin18° (exact) =
A
(√5−1)/4
B
(√5+1)/4
C
(√5−1)/2
D
1/4
70
cos36° (exact) =
A
(√5−1)/4
B
(√5+1)/4
C
(1+√5)/4
D
√5/4
71
If A+B+C=π (triangle), then sinA+sinB+sinC =
A
4cosA/2 cosB/2 cosC/2
B
sinAsinBsinC
C
4sinA/2sinB/2sinC/2
D
sinA+sinB+sinC (cannot simplify)
72
tanA × cot A =
A
0
B
1
C
tan²A
D
2
73
cos4A in terms of cos2A:
A
2cos²2A−1
B
1−2cos²2A
C
2cos²A−1
D
cos²2A
74
sin 2A = 2sin A cos A. For A=30°:
A
sin60°=√3/2=2(1/2)(√3/2)✓
B
sin60°=√3
C
Not equal
D
1
75
3sinA − 4sin³A = sin3A. For A=30°: LHS=
A
3/2−4/8=3/2−1/2=1
B
1
C
sin90°=1
D
All
76
sec²A − tan²A =
A
0
B
1
C
sec²A
D
tan²A
77
csc²A − cot²A =
A
0
B
1
C
csc²A
D
cot²A
78
sin(A+B)−sin(A−B) =
A
2sinAcosB
B
2cosAsinB
C
2sinAsinB
D
2cosAcosB
79
cos(A+B)+cos(A−B) =
A
2cosAcosB
B
2sinAsinB
C
2sinAcosB
D
2cosAsinB
80
For acute A: if sin A = 5/13, cos A =
A
12/13
B
5/12
C
13/5
D
5/13
81
For that A: tan A =
A
5/12
B
12/13
C
5/13
D
12/5
82
Verify sin(π/2−A)=cosA for A=π/3: sin(π/2−π/3)=sin(π/6)=1/2; cos(π/3)=
A
1/2
B
√3/2
C
1
D
0
83
sin2A × cos2A = ?
A
sin4A/2
B
sin4A
C
2sin2Acos2A
D
sinAcosA
0
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