Download Practice Test 3 - Lemon Bay High School

PRACTICE TEST 3
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SHOW YOUR WORK.
Establish the identity.
1) cot · sec = csc
1)
2) (1 - cos x)(1 + cos x) = sin2 x
2)
3)
tan u - 1 1 - cot u
=
tan u + 1 1 + cot u
3)
4)
1 + csc x
= cos x + cot x
sec x
4)
5)
sin x
sin x
+
= 2 csc x
1 - cos x 1 + cos x
5)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the exact value of the expression.
11
6) sin 12
A)
7) sin
6
B) -
6+
4
2
C)
2+
4
6
D)
64
2
7)
12
A)
8) tan
24
6)
2( 3 - 1)
B)
2( 3 - 1)
4
C) - 2( 3 - 1)
2( 3 - 1)
4
D) -
13
12
8)
A) 2 +
3
B) -2 -
3
C)
3-2
D) 2 -
3
9) sin 15°
A)
9)
2( 3 + 1)
4
10) tan 75°
A) - 3 - 2
B) -
2( 3 + 1)
4
2( 3 - 1)
4
C) -
D)
2( 3 - 1)
4
10)
B)
3+2
C)
11) sin 185° cos 65° - cos 185° sin 65°
3
1
A) B) 2
2
3-2
D) - 3 + 2
11)
37
C)
12
1
3
D)
2
12) sin 15° cos 105° + cos 15° sin 105°
3
1
A) B) 2
2
12)
3
2
C)
D)
1
4
13) cos 35° cos 25° - sin 35° sin 25°
3
A)
14) cos
2
2
14)
B) -1
C) 0
D) 1
tan 5° + tan 25°
1 - tan 5° tan 25°
A) 2
16)
1
D)
2
5
2
5
2
cos
sin
- sin
18
9
18
9
A)
15)
13)
3
C)
2
1
B)
4
15)
3
B)
C)
1
2
3
3
D)
tan 40° + tan 110°
1 - tan 40° tan 110°
A) -
1
2
16)
B) -
3
3
C) - 3
D) -2
1
3
17) sin cos-1 - sin-1
2
2
A)
2 3
2
18) cos tan-1
17)
B) 0
C)
3
3
D) 1
4
3
- sin-1
3
5
A) 1
18)
B)
2 3
5
C)
24
25
D)
2 6
5
2
1
19) sin sin-1 + cos-1
3
3
A)
2 + 2 10
9
20) tan tan-1
A)
19)
B)
2 3
5
C)
2 3 + 2 10
9
D)
2 6
5
3
1
+ sin-1
4
2
9+4 3
12 - 3 3
20)
B)
2 3
5
C)
2
2 3 + 2 10
9
D)
2 6
5
Use the information given about the angle , 0
2 , to find the exact value of the indicated trigonometric function.
5
, 0< <
Find cos(2 ).
21) sin =
21)
13
2
A)
22) cos
119
169
=
A) -
B)
3 3
,
<
5
2
24
,
7
A) -
336
625
24) csc
=-
A) -
25) sec
A)
26) sin
A)
27) sin
A)
<2
B) -
<
<
3
2
3
<
2
D)
7
25
C)
336
625
D) -
<2
7
25
24)
C)
4 21
25
D)
17
25
25)
C) -
17
25
D)
4 21
25
26)
7
25
C) -
24
25
D)
24
25
Find sin(2 ).
B) -
27)
24
25
C)
24
25
D) -
Find the exact value of the expression.
3
28) sin 2 cos-1
2
A)
3
527
625
Find cos(2 ).
B) -
4
,
5
24
25
23)
-4 21
25
<2
7
25
=-
C)
Find sin(2 ).
B)
4 3
,
<
5
2
24
25
-4 21
25
>0
17
25
=-
9
13
Find cos(2 ).
B)
5 21
, csc
21
D)
22)
527
625
>0
17
25
=-
119
169
Find sin(2 ).
B)
5
, tan
2
C) -
Find sin(2 ).
7
25
=
23) tan
120
169
7
25
28)
B) -
3
2
C)
3
3
2
D)
1
2
29) sin 2 sin-1
2
2
29)
A) 0
B) 1
30) cos 2 sin-1 A)
3
C)
D)
1
2
5
13
30)
119
169
B)
10
13
C) -
12
13
D)
2 5 + 10
13
12
31) cos 2 tan-1
5
31)
3
13
A) -
B) -
119
169
C)
10
13
D) -
144
169
Use the information given about the angle , 0
2 , to find the exact value of the indicated trigonometric function.
1
Find sin .
32) sin = , 0 < <
32)
4
2
2
8 + 2 15
4
A)
33) sin
1
, tan
4
=
A)
35) cos
A)
>0
8 + 2 15
4
A)
34) csc
B)
=-
>0
5
6
=
6
4
<
B)
8+2
4
10
4
D)
6
4
34)
18 - 6 5
6
3+
6
D) -
5
.
35)
15
C)
8-2
4
15
Use the Half-angle Formulas to find the exact value of the trigonometric function.
36) sin 22.5°
1
1
1
2+ 2
2+ 2
2- 2
A)
B) C)
2
2
2
37) tan 165°
A) -2 +
10
4
.
C) -
2
D)
33)
C)
2
8 - 2 15
4
.
18 + 6 5
6
Find cos
2
2
8 - 2 15
4
Find cos
B)
1
, 0<
4
C)
Find cos
B)
3
, tan
2
6
4
D)
10
4
1
D) 2
36)
2-
2
37)
3
B) -2 -
3
C) 2 +
4
3
D) 2 -
3
38) tan 75°
A) -2 +
39) sin
3
B) -2 -
3
C) 2 +
3
D) 2 -
3
39)
12
A)
40) tan
38)
1
2
1-
3
B)
1
2
1-
3
C)
1
2
2+
3
D)
1
2
2-
40)
8
A) 1 +
2
B) -1 -
2
C) -1 +
2
D) 1 -
2
Express the product as a sum containing only sines or cosines.
41) sin(4 ) cos(2 )
1
1
A) [cos(6 ) - cos(2 )]
B) [sin(6 ) + cos(2 )]
2
2
C) sin cos(8 2 )
D)
42) sin(7 ) sin(4 )
1
[cos(11 ) - cos(3 )]
2
D)
43) cos(6 ) cos(5 )
1
A) [cos(11 ) - cos ]
2
C)
1
[ cos
2
1
[sin(6 ) + sin(2 )]
2
42)
1
[sin(11 ) + cos(3 )]
2
43)
B) cos2 (30 2 )
+ cos(11 )]
D)
44) sin(3 ) sin(7 )
1
[cos(11 ) - sin ]
2
1
B) [- cos(4 ) - cos(10 )]
2
A) sin2 (21 2 )
C)
41)
1
B) [ cos(3 ) - cos(11 )]
2
A) sin2 (28 2 )
C)
3
1
[cos(4 ) - cos(10 )]
2
D)
1
[cos(10 ) - sin(4 )]
2
Express the sum or difference as a product of sines and/or cosines.
45) sin(7 ) + sin(3 )
A) 2 cos(5 ) sin(2 )
B) 2 sin(5 ) cos(2 )
C) 2 sin(5 ) sin(2 )
D) 2 sin(10 )
46) cos(5 ) + cos(3 )
A) 2 sin(4 ) sin
B) 2 cos(4 ) cos
C) 2 cos(4 ) sin
5
44)
45)
D) 2 cos(4 )
46)
47) sin(4 ) - sin(6 )
A) 2 cos(4 ) cos(5 )
C) 2 sin(5 ) cos
48) cos
B) -2 sin
D) -2 sin
47)
cos(5 )
9
7
+ cos
2
2
A) 2 cos(4 )
49) sin(4 ) - sin(2 )
A) sin cos(3 )
48)
B) 2 cos(4 ) cos
2
B) 2 sin(3 ) cos
C) 2 sin(4 ) sin
D) 2 sin(4 ) sin
C) sin(3 ) cos
D) 2 sin
2
cos(3 )
Solve the problem.
50) On a Touch-Tone phone, each button produces a unique sound. The sound produced is the sum of
two tones, given by
y = sin (2 lt) and y = sin (2 ht)
where l and h are the low and high frequencies (cycles per second) shown on the illustration.
49)
50)
The sound produced is thus given by
y = sin (2 lt) + sin (2 ht)
Write the sound emitted by touching the 6 key as a product of sines and cosines.
A) y = 2 sin(439 t) cos(1979 t)
B) y = 2 sin(2247 t) cos(707 t)
C) y = 2 sin(707 t) cos(2247 t)
D) y = 2 sin(1979 t) cos(439 t)
51) A surveyor is measuring the distance across a small lake. He has set up his transit on one side of the
lake 90 feet from a piling that is directly across from a pier on the other side of the lake. From his
transit, the angle between the piling and the pier is 40°. What is the distance between the piling and
the pier to the nearest foot?
A) 58 ft
B) 107 ft
C) 76 ft
D) 69 ft
51)
52) A radio transmission tower is 240 feet tall. How long should a guy wire be if it is to be attached 15
feet from the top and is to make an angle of 20° with the ground? Give your answer to the nearest
tenth of a foot.
A) 239.4 ft
B) 657.9 ft
C) 701.7 ft
D) 255.4 ft
52)
53) A building 260 feet tall casts a 100 foot long shadow. If a person looks down from the top of the
building, what is the measure of the angle between the end of the shadow and the vertical side of
the building (to the nearest degree)? (Assume the person's eyes are level with the top of the
building.)
A) 21°
B) 69°
C) 23°
D) 67°
53)
6
54) John (whose line of sight is 6 ft above horizontal) is trying to estimate the height of a tall oak tree.
He first measures the angle of elevation from where he is standing as 35°. He walks 30 feet closer to
the tree and finds that the angle of elevation has increased by 12°. Estimate the height of the tree
rounded to the nearest whole number.
A) 90 ft
B) 61 ft
C) 86 ft
D) 67 ft
Solve the triangle.
55)
54)
55)
25°
125°
4
A) C = 30°, a = 4.73, c = 7.75
C) C = 30°, a = 7.75, c = 4.73
B) C = 25°, a = 4.73, c = 7.75
D) C = 35°, a = 7.75, c = 4.73
56) A = 20°, B = 10°, a = 1
A) C = 150°, b = 1.46, c = 0.51
C) C = 150°, b = 1.46, c = -0.49
B) C = 150°, b = 0.51, c = 1.46
D) C = 150°, b = 1.51, c = 1.46
56)
Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no
triangle at all. Solve any triangle(s) that results.
57) a = 7, b = 9, B = 49°
57)
A) one triangle
B) two triangles
A = 35.94°, C = 95.06°, c = 11.88
A1 = 76.01°, C1 = 54.99°, c1 = 7.60 or
A2 = 103.99°, C2 = 27.01, c2 = 12.14
D) no triangle
C) one triangle
A = 76.01°, C = 54.99°, c = 7.60
58) b = 3, c = 7, B = 80°
A) one triangle
C = 41°, A = 59°, a = 14
C) one triangle
B = 40°, A = 60°, a = 10
B) one triangle
C = 39°, A = 61°, a = 12
D) no triangle
59) a = 10, b = 7, B = 10°
A) one triangle
A = 165.64°, C = 4.36°, c = 3.06
B) two triangles
A1 = 14.36°, C1 = 155.64°, c1 = 16.63 or
58)
59)
A2 = 165.64°, C2 = 4.36°, c2 = 3.06
D) no triangle
C) one triangle
A = 14.36°, C = 155.64°, c = 16.63
60) A = 30°, a = 14, b = 28
A) B = 60°, C = 60°, c = 24.2
C) B = 90°, C = 60°, c = 24.2
B) B = 60°, C = 90°, c = 24.2
D) no triangle
7
60)
61) A = 65°, a = 5, b = 7
A) one triangle
B = 32°, C = 83°, c = 14
C) one triangle
A = 33°, C = 82°, c = 12
B) one triangle
B = 34°, C = 81°, c = 16
D) no triangle
62) C = 35°, a = 18.7, c = 16.1
A) two triangles
A1 = 42°, B1 = 103°, b1 = 27.4;
B) one triangle
A = 42°, B = 103°, b = 27.4
A2 = 138°, B2 = 7°, b2 = 3.4
C) two triangles
A1 = 103°, B1 = 42°, b1 = 27.4;
61)
62)
D) no triangle
A2 = 7°, B2 = 138°, b2 = 3.4
Solve the problem.
63) A ship at sea, the Admiral, spots two other ships, the Barstow and the Cauldrew and measures the
angle between them at be 45°. They radio the Barstow and by comparing known landmarks, the
distance between the the Admiral and the Barstow is found to be 323 meters. The Barstow reports
an angle of 59° between the Admiral and the Cauldrew. To the nearest meter, what is the distance
between the Barstow and the Cauldrew?
A) 49 m
B) 81 m
C) 235 m
D) 266 m
64) It is 4.7 km from Lighthouse A to Port B. The bearing of the port from the lighthouse is N73°E. A
ship has sailed due west from the port and its bearing from the lighthouse is N31°E. How far has
the ship sailed from the port? Round your answer to the nearest 0.1 km.
A) 2.7 km
B) 3.1 km
C) 3.5 km
D) 3.7 km
Solve the triangle.
65)
63)
64)
65)
3
110°
4
A) c = 5.76, A = 29.3°, B = 40.7°
C) c = 4.76, A =40.7°, B = 29.3°
B) c = 6.76, A = 29.3°, B = 40.7°
D) c = 5.76, A = 40.7°, B = 29.3°
66) b = 4, c = 5, A = 95°
A) a = 6.67, B = 48.3°, C = 36.7°
C) a = 7.67, B = 36.7°, C = 48.3°
B) a = 5.67, B = 48.3°, C = 36.7°
D) a = 6.67, B = 36.7°, C = 48.3°
8
66)
67)
67)
9
6
4
A) A = 127.2°, B = 32.1°, C = 20.7°
C) A = 127.2°, B = 20.7°, C = 32.1°
B) A = 32.1°, B = 20.7°, C = 127.2°
D) A = 32.1°, B = 127.2°, C = 20.7°
68) a = 5, b = 5, c = 3
A) A = 72.5°, B = 35°, C = 72.5°
C) A = 35°, B = 72.5°, C = 72.5°
B) A = 73.5°, B = 73.5°, C = 33°
D) A = 72.5°, B = 72.5°, C = 35°
Solve the problem.
69) In flying the 81 miles from Champaign to Peoria, a student pilot sets a heading that is 8° off course
and maintains an average speed of 88 miles per hour. After 15 minutes, the instructor notices the
course error and tells the student to correct his heading. Through what angle will the plane move to
correct the heading and how many miles away is Peoria when the plane turns?
A) 169°; 72.67 mi
B) 11°; 59.29 mi
C) 11°; 72.67 mi
D) 169°; 59.29 mi
68)
69)
70) Two points A and B are on opposite sides of a building. A surveyor selects a third point C to place a
transit. Point C is 50 feet from point A and 61 feet from point B. The angle ACB is 57°. How far
apart are points A and B?
A) 88.8 ft
B) 67.5 ft
C) 97.7 ft
D) 53.8 ft
70)
71) The distance from home plate to dead center field in a certain baseball stadium is 410 feet. A
baseball diamond is a square with a distance from home plate to first base of 90 feet. How far is it
from first base to dead center field?
A) 477.9 ft
B) 335.1 ft
C) 352.2 ft
D) 387.4 ft
71)
9
Answer Key
Testname: PRACTICETEST3
1) cot
· sec
=
cos
sin
·
1
cos
=
1
sin
= csc
2) (1 - cos x)(1 + cos x) = 1 - cos2 x = sin2 x
1
1 - cot u
-1
cot
u
cot u
tan u - 1
1 - cot u
3)
=
=
=
tan u + 1
1
1 + cot u
1 + cot u
+1
cot u
cot u
4)
1 + csc x
1
cos x (sin x + 1) cos x sin x cos x
= cos x 1 +
=
=
+
= cos x + cot x.
sec x
sin x
sinx
sin x
sin x
5)
sin x
sin x
sin x[(1 + cos x)+(1 - cos x)]
2 sin x
2 sin x
+
=
=
=
= 2 csc x.
1 - cos x 1 + cos x
(1 - cos x)(1 + cos x)
2
1 - cos x
sin2 x
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
23)
24)
25)
26)
27)
28)
29)
30)
31)
32)
33)
34)
35)
36)
37)
38)
39)
40)
41)
42)
A
B
D
D
B
D
C
D
C
D
B
B
C
A
A
A
B
C
D
B
B
B
C
B
A
B
C
A
C
D
C
A
C
D
D
D
B
10
Answer Key
Testname: PRACTICETEST3
43)
44)
45)
46)
47)
48)
49)
50)
51)
52)
53)
54)
55)
56)
57)
58)
59)
60)
61)
62)
63)
64)
65)
66)
67)
68)
69)
70)
71)
C
C
B
B
B
B
D
B
C
B
A
D
C
B
A
D
B
C
D
A
C
D
A
D
A
D
B
D
C
11