Download 1 FORMULAS SHEET: MATH 1112 Fundamental Identities

DR. YOU: TRIGONOMETRY
FORMULAS SHEET: MATH 1112
Fundamental
tan 𝜃 =
Identities
Half angle
Formulas
sin 𝜃
cos 𝜃
cot 𝜃 =
cos 𝜃
sin 𝜃
cot 𝜃 =
1
tan 𝜃
csc 𝜃 =
sin2 𝜃 + cos 2 𝜃 = 1
tan2 𝜃 + 1 = sec 2 𝜃
𝜃
1 − cos 𝜃
sin ( ) = ±√
2
2
𝜃
1 + cos 𝜃
cos ( ) = ±√
2
2
1
sin 𝜃
sec 𝜃 =
1
cos 𝜃
cot 2 𝜃 + 1 = csc 2 𝜃
𝜃
1 − cos 𝜃
tan ( ) =
2
sin 𝜃
𝜃
Formulas
( the sign is determined by the quadrant of 2 )
2 tan 𝜃
sin(2𝜃) = 2 sin 𝜃 cos 𝜃
tan(2𝜃) =
1 − tan2 𝜃
Sum and
sin(𝑎 + 𝑏) = sin 𝑎 cos 𝑏 + cos 𝑎 sin 𝑏
difference
sin(𝑎 − 𝑏) = sin 𝑎 cos 𝑏 − cos 𝑎 sin 𝑏
Formulas
cos(𝑎 + 𝑏) = cos 𝑎 cos 𝑏 − sin 𝑎 sin 𝑏
Double angle
cos(2𝜃) = cos 2 𝜃 − sin2 𝜃
cos(2𝜃) = 2 cos2 𝜃 − 1
cos(𝑎 − 𝑏) = cos 𝑎 cos 𝑏 + sin 𝑎 sin 𝑏
Sum to product
Formulas
Product to Sum
Formulas
Solving
𝑎+𝑏
𝑎−𝑏
) cos (
)
2
2
𝑎−𝑏
𝑎+𝑏
sin 𝑎 − sin 𝑏 = 2 sin (
) cos (
)
2
2
1
sin 𝑎 sin 𝑏 = {cos(𝑎 − 𝑏) − cos(𝑎 + 𝑏)}
2
1
sin 𝑎 cos 𝑏 = {sin(𝑎 + 𝑏) + sin(𝑎 − 𝑏)}
2
sin 𝑎 + sin 𝑏 = 2 sin (
1
2
cos(2𝜃) = 1 − 2 sin2 𝜃
tan(𝑎 + 𝑏) =
tan 𝑎 + tan 𝑏
1 − tan 𝑎 tan 𝑏
tan(𝑎 − 𝑏) =
tan 𝑎 − tan 𝑏
1 + tan 𝑎 tan 𝑏
𝑎+𝑏
𝑎−𝑏
cos 𝑎 + cos 𝑏 = 2 cos (
) cos (
)
2
2
𝑎+𝑏
𝑎−𝑏
cos 𝑎 − cos 𝑏 = −2 sin (
) sin (
)
2
2
1
cos 𝑎 cos 𝑏 = {cos(𝑎 − 𝑏) + cos(𝑎 + 𝑏)}
2
Area = 𝑎𝑏 sin(𝐶) or
Area = √𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐), 𝑠 =
LAW OF Sines
LAW of Cosines
Triangles
sin(𝐴) sin(𝐵) sin(𝐶)
=
=
𝑎
𝑏
𝑐
Complex
numbers
𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 cos(𝐶)
Let 𝑧1 = 𝑟1 (cos(𝑎) + 𝑖 sin(𝑎)) and 𝑧2 = 𝑟2 (cos(𝑏) + 𝑖 sin(𝑏))
□ 𝑧1 𝑧2 = 𝑟1 𝑟2 (cos(𝑎 + 𝑏) + 𝑖 sin(𝑎 + 𝑏))
□
𝑧1
𝑧2
𝑟
= 𝑟1 (cos(𝑎 − 𝑏) + 𝑖 sin(𝑎 − 𝑏))
2
Let 𝑧 = 𝑎 + 𝑏𝑖 be a complex number
□ The conjugate of 𝑧 is 𝑧̅ = 𝑎 − 𝑏𝑖
□ |𝑧| = √𝑎2 + 𝑏 2
Vector
Let 𝐯 = 𝑎1 𝐢 + 𝑏1 𝐣 and 𝐰 = 𝑎2 𝐢 + 𝑏2 𝐣
𝐯
□ Unit vector in same direction as 𝑣 is 𝐮 = ‖𝐯‖ , ‖𝐯‖ = √𝑎1 2 + 𝑏1 2
□ Dot product: 𝐯  𝐰 = 𝑎1 𝑎2 + 𝑏1 𝑏2
𝐯𝐰
□ If 𝜃 is between vectors 𝑣 and 𝑤:cos 𝜃 = ‖𝐯‖∙‖𝐰‖
𝑎+𝑏+𝑐
2
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DR. YOU: TRIGONOMETRY
PRACTICE PROBLEMS FOR TRIGONOMETRY FINAL
1. Convert 84° to exact radian measure.
(A)
11𝜋
15
2. Convert
(B)
13𝜋
12
7𝜋
15
(C)
7𝜋
6
7𝜋
7
12
(D) 12
(E)
(D) 225°
(E) 235°
to exact degree measure.
(A) 105°
(B) 135°
(C) 195°
20𝜋
).
3
3. Evaluate cos (
(A)
√3
2
(B) −
√3
2
1
1
2𝜋
3
(D) − 2
(E)
(D) −1.7710
(E) −0.8253
(D) −0.2958
(E) None of above
(C) −0.7°
(D) −67.6°
(E) None of above
(C) 66.4°
(D) 21.8°
(E) None of above
(C) 2
4. Use a calculator to evaluate csc(−0.6). Round to 4 decimal places.
(A) 1.2116
(B) 1.0001
(C) −1.2116
5. Find the approximate radian value for cos −1(−0.2915).
(A) 0.2874
(B) −1.2750
(C) 1.8666
6. Find the approximate degree for sin−1 (−0.6257).
(A) −38.7°
(B) 51.3°
7. Find the approximate degree for sec −1(2.5).
(A) 0.5°
(B) 23.6°
8. Tom is riding a bike with 14-inch diameter wheels which are rotating 6 revolutions per second. What is his linear
speed in feet per minute?
(A) 527.79
(B) 659.73
(C) 1319.47
(D) 2638.94
(E) None of above
9. Find the length of the arc subtended by a central angle of 135° on a circle of radius 4 meters.
(A) 6𝜋 m
(B) 36𝜋 m
(C) 540 m
(D) 1.5𝜋 m
(E) 3𝜋 m
10. A child is spinning a rock at the end of a 2-foot rope at the rate of 180 revolutions per minute (rpm). Find the angular
speed of the rock in radian per minute.
(A) 180𝜋
(B) 360𝜋
(C)
𝜋
2
(D) 2𝜋
(E) 𝜋
11. Find the area of a sector a circle with the radius of 10 cm and a central angel of 120°. Round to the nearest hundredth
of a square centimeter.
(A) 600.00
(B) 104.72
(C) 209.44
(D) 10.92
(E) 657.97
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DR. YOU: TRIGONOMETRY
12. A car is traveling 55 mph. If the radius of the wheel is 11 inches, find the angular velocity in revolutions per minute.
(A) 600 rpm
(B) 695 rpm
(C) 735 rpm
(D) 840 rpm
(E) 1200 rpm
13. The graph shown below is the graph of which trigonometric function over the domain from −2𝜋 ≤ 𝑥 ≤ 2𝜋 with 𝑥axis scale of
𝜋
2
and 𝑦-axis scale of 1.
(A) 𝑦 = cos 𝑥
4
(B) 𝑦 = sec 𝑥
2
π
2π
π
(C) 𝑦 = cot 𝑥
2π
(D) 𝑦 = csc 𝑥
2
4
𝜋
2
14. Graph the function 𝑦 = 5 sin (2𝑥 − ) over the domain from [−2𝜋, 2𝜋] with 𝑥-axis scale of
(A)
4
2
π
and 𝑦-axis scale of 1.
(B)
4
2π
𝜋
2
2
π
2π
π
2π
2
π
2π
π
2π
2
4
4
(C)
(D)
4
4
2
2
2π
π
π
π
2π
2π
2
2
4
4
Instructions for questions 14-16. Find the values of the indicated trigonometric functions of the angle 𝜃, which is in
standard position with the terminal side passing through the point 𝑃(12, −5).
15. Find the value of the sine function, sin 𝜃, using the point 𝑃 listed above.
5
(A) − 13
5
(B) 12
12
(C) − 13
12
(E) − 12
12
(E) − 12
12
5
(E) −
(D) 13
5
16. Find the value of the cosine function, cos 𝜃, using the point 𝑃 listed above.
5
(A) − 13
5
(B) 12
12
(C) − 13
(D) 13
5
17. Find the value of the cotangent function, cot 𝜃, using the point 𝑃 listed above.
13
(A) − 12
5
(B) 12
5
(C) − 12
(D)
12
5
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DR. YOU: TRIGONOMETRY
18. If the function 𝑦 = cot(𝑥) is graphed over the domain −2𝜋 ≤ 𝑥 ≤ 2𝜋, this graph has vertical asymptotes (VA) and 𝑥intercepts at the following:
(A) VA at −2𝜋, −𝜋, 0 𝜋, 2𝜋 and 𝑥-intercepts at −
(B) VA at −
3𝜋
𝜋 𝜋 3𝜋
,−2,2, 2
2
(C) VA at −
3𝜋
𝜋
, − 4,
4
(D) VA at −
3𝜋
𝜋 𝜋 3𝜋
,−2,2, 2
2
3𝜋
𝜋 𝜋 3𝜋
,−2,2, 2
2
and 𝑥-intercepts at −2𝜋, −𝜋, 0, 𝜋, 2𝜋
𝜋 3𝜋
, 4
4
and 𝑥-intercepts at −
and 𝑥-intercepts at −
3𝜋
𝜋 𝜋 3𝜋
,−2,2, 2
2
3𝜋
𝜋
, − 4,
4
𝜋 3𝜋
, 4
4
𝑥
2
19. State the amplitude(A) and period(P) of the graph of 𝑓(𝑥) = 3 sin ( ).
(A) 𝐴 = 3 ; 𝑃 =
(D) 𝐴 =
1
2
𝜋
2
(B) 𝐴 = 3 ; 𝑃 = 𝜋
(C) 𝐴 = 3 ; 𝑃 = 4𝜋
(E) None of these
;𝑃 = 3
20. State the period of the function 𝑓(𝑥) = −2 tan(4𝑥).
(A) 𝜋
(B) 2𝜋
(C) 4𝜋
(D)
𝜋
2
(E)
𝜋
4
(E)
3
4
4
5
21. If 𝜃 is an acute angle in the standard position and cos 𝜃 = , find the value of sin 𝜃.
4
4
(A) 3
22. Given that sin 𝜃 = −
(A)
√3
3
√3
2
3
5
(B) 5
(D) 5
(C) 3
for an angle in quadrant IV, find the exact value of cot 𝜃.
(B) −
√3
3
(C) √3
(D) −√3
1
(E) − 2
1
23. Find the exact value of cos−1 (− 2).
(A) −
𝜋
3
(B) −
𝜋
6
(C)
𝜋
6
(D)
𝜋
3
(E)
2𝜋
3
(D)
12
5
(E)
13
12
(E)
3
4
(E)
7𝜋
5
5
24. Find the exact value of sin (cos −1 (13)).
(A) 67.4°
12
5
(B) 13
(C) 12
3
25. Find the exact value of cos (tan−1 (− 4))
4
(A) − 5
4
3
(C) − 5
(B) 5
26. Find the exact value of sin−1 (sin (−
(A)
2𝜋
5
(B) −
2𝜋
5
3
(D) 5
3𝜋
))
5
(C)
3𝜋
5
(D) −
3𝜋
5
4
DR. YOU: TRIGONOMETRY
27. Write cos(2𝑥) cos(3𝑥) − sin(2𝑥) sin(3𝑥) in terms of a single trigonometric function.
(A) sin(−𝑥)
(B) sin(5𝑥)
(C) cos(𝑥)
3
5
(D) cos(5𝑥)
(E) None of these
𝜋
2
28. Find the exact value of cos(2𝑥) if sin 𝑥 = and 0 < 𝑥 < .
(A) −
4
5
(B)
7
25
(C) −
4
5
29. Given cos 𝛼 = − with 𝛼 in quadrant II and sin 𝛽 = −
56
(B)
(A) 65
36
25
18
25
5
13
(D)
(E) −
7
25
with 𝛽 in quadrant IV, find the exact value of sin(𝛼 + 𝛽).
36
65
(E) None of these
(D) − 65
33
(E) None of these
(D) 2 − √2
(E) None of these
56
(D) −
(C) − 65
63
(C) √2 − 1
(C) − 65
4
18
25
12
30. Find the exact value of sin [cos−1 (5) + sin−1 (− 13)].
63
33
(A) 65
(B) 65
31. Use a half-angle formula to evaluate tan(22.5°).
(A) 2√2 − 2
(B) √2 + 2
32. Use a half-angle formula to evaluate sin(15°)
(A)
√2−√2
(B)
2
√2+√2
2
(C)
√2−√3
(D)
2
√2+√3
(E) None of these
2
33. Find the length of the indicated side, 𝑥, in the right triangle below;
(A) 10.064 cm
57°
12 cm
(B) 14.308 cm
(C) 6.537 cm
(D) 22.033 cm
x cm
(E) 18.478 cm
34. A 22-foot extension ladder leaning against a building makes a 70° angle with the ground. How far up the building
does the ladder touch?
(A) 17.026 ft
(B) 20.67 ft
(C) 23.41 ft
(D) 6104.89 ft
(E) None of these
35. To find the distance across a canyon from point B to point C, a surveying team locates points A and B on one side of
the canyon and point C on the other side of the canyon. The distance between A and B is 92 yards. Angle CAB
measures 67°, and angle CBA is 89°. Find the distance across the canyon from point B to point C. Round to the
nearest yard.
(A) 150 yd
(B) 208 yd
(C) 230 yd
(D) 248 yd
(E) 350 yd
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DR. YOU: TRIGONOMETRY
36. The measure of the angle of elevation from a position 65 feet from the base to the top of a flagpole is 32°. Find the
height of the flagpole to the nearest tenth of a foot.
(A) 34.4 ft
(B) 40.6 ft
(C) 55.1 ft
(D) 72.4 ft
(E) 104.0 ft
37. Find the measure of angle 𝐴 of a right triangle if 𝑐 = 25 cm, 𝐶 = 90°, and 𝑏 = 14 cm
(A) 29°
(B) 34°
(C) 56°
(D) 61°
(E) None of these
38. Find the measure of angle 𝐵 in a triangle ABC if 𝑎 = 24 m, 𝑏 = 47m and 𝐴 = 36°.
(A) 18°
(B) 18°, 163°
(C) 60°
(D) 54°
(E) None of these
39. In a triangle ABC, 𝑎 = 10 km, 𝑏 = 20 km, and 𝐶 = 110°. Find the length of the side 𝑐.
(A) 20.8 km
(B) 19.1 km
(C) 25.2 km
(D) 23.8 km
(E) None of these
40. In a triangle ABC, 𝑎 = 21 ft, 𝑏 = 19 ft, and 𝑐 = 25 ft. Find the measure of the angle 𝐵.
(A) 48°
(B) 42°
(C) 55°
(D) 77°
(E) 65°
41. In a triangle ABC, 𝑎 = 10 m, 𝑏 = 6 m, and 𝐶 = 61°. Find the area of the triangle ABC. Round to the nearest square
meter.
(A) 15 m2
(B) 26 m2
(C) 30 m2
(D) 52 m2
(E) 105 m2
42. In a triangle ABC, 𝑎 = 24 ft, 𝑏 = 28 ft, and 𝑐 = 30 ft. Find the area of the triangle ABC. Round to the nearest square
feet.
(A) 285 ft 2
(B) 316 ft 2
(C) 336 ft 2
(D) 909 ft 2
(E) 99671 ft 2
43. Two observation points A and B (which are left side of an airplane) are 950 ft apart. From these points the angle of
elevation of an airplane are 52° and 57°, respectively. Find the height of the airplane.
(A) 1462.87 ft
(B) 1215.94 ft
(C) 7000.98 ft
(D) 7203.63 ft
(E) 9500 ft
44. A boat travels at 55 mph for one hour at a bearing of 315°. Then the boat travels at 50 mph for 2 hours at a bearing of
225°. At the end of these 3 hours, how far is the boat form the starting point? Round to the nearest mile.
(A) 74 miles
(B) 90 miles
(C) 105 miles
45. The polar coordinates of a point are given as (−5,
5√2 5√2
, 2 )
2
(A) (
(B) (−
5√2 5√2
, 2 )
2
7𝜋
).
4
(D) 110 miles
(E) 114 miles
Find the rectangular coordinates for this point.
5√2
5√2
,− 2 )
2
(C) (
(D) (−
5√2
5√2
,− 2 )
2
5 5√3
)
2
(E) (2 ,
6
DR. YOU: TRIGONOMETRY
46. The rectangular coordinates of a given point are given as (1, −√3). Find an equivalent pair of the polar coordinates.
𝜋
(A) (2, 3 )
(B) (2,
2𝜋
)
3
(C) (2,
4𝜋
)
3
(D) (2,
5𝜋
)
3
2𝜋
)
3
(E) (2, −
47. Write 𝑧 = −1 − 𝑖√3 in polar form.
4𝜋
4𝜋
(A) √3 [cos ( 3 ) + 𝑖 sin ( 3 )]
𝜋
𝜋
(D) √3 [cos ( 3 ) + 𝑖 sin ( 3 )]
𝜋
𝜋
(B) 2 [cos ( 3 ) + 𝑖 sin ( 3 )]
7𝜋
4𝜋
4𝜋
(C) 2 [cos ( 3 ) + 𝑖 sin ( 3 )]
7𝜋
(E) 2 [cos ( 6 ) + 𝑖 sin ( 6 )]
48. Write 𝑧 = 6[cos(60°) + 𝑖 sin(60°)] in the standard form 𝑥 + 𝑖𝑦.
(A) 3 + 3√3𝑖
(B) 3√3 + 3𝑖
(C) 6 + 6√3𝑖
(E) None of these
(D) 6√3 + 6𝑖
49. Find the product 𝑧𝑤 of 𝑧 = 2[cos(155°) + 𝑖 sin(155°)] and 𝑤 = 6[cos(275°) + 𝑖 sin(275°)]
(A) 8[cos(130°) + 𝑖 sin(130°)]
(B) 12[cos(70°) + 𝑖 sin(70°)]
(D) 12[cos(120°) + 𝑖 sin(120°)]
(E) 3[cos(120°) + 𝑖 sin(120°)]
5𝜋
(C) 12[cos(265°) + 𝑖 sin(265°)]
5𝜋
50. Write {2 [cos ( 16 ) + 𝑖 sin ( 16 )]} 4 in the standard form 𝑥 + 𝑖𝑦.
5𝜋
5𝜋
(B) 2 [cos ( 4 ) + 𝑖 sin ( 4 )]
9𝜋
16
9𝜋
16
(E) 16 [cos ( ) + 𝑖 sin ( )]
(A) 8 [cos ( 4 ) + 𝑖 sin ( 4 )]
(D) 8 [cos ( ) + 𝑖 sin ( )]
5𝜋
9𝜋
16
5𝜋
5𝜋
5𝜋
(C) 16 [cos ( 4 ) + 𝑖 sin ( 4 )]
9𝜋
16
8
51. Write (1 + √3 𝑖) in the standard form 𝑥 + 𝑖𝑦.
(A) 128 − 128√3 𝑖
(B) −4 + 4√3 𝑖
(D) 4 − 4√3 𝑖
(E) None of these
(C) −128 + 128√3 𝑖
52. Convert 36°9′ to decimal degree.
(A) 36.1°
(B) 36.9°
(C) 36.15°
(D) 36.05°
(E) 36.8°
53. Given the vector, ⃗⃗⃗⃗⃗
𝑃𝑄 , whose initial point is 𝑃 = (−2,4) and whose terminal point is 𝑄 = (4, −3), the position vector,
⃗⃗⃗⃗⃗ , is given by
v, which is equal to vector, 𝑃𝑄
(A) 6𝐢 + 𝐣
(B) 6𝐢 − 7𝐣
(C) 2𝐢 + 𝐣
(D) −3𝐢 − 𝐣
(E) −6𝐢 + 7𝐣
54. If 𝜃 is the angle between the two vectors, v and w, find the measure of angle 𝜃 in degrees, where v = 2𝐢 − 3𝐣 and
w = 4𝐢 + 2𝐣. Round to the nearest tenth.
(A) 82.9°
(B) 29.7°
(C) 3.1°
(D) 60.3°
(E) 71.9°
7
DR. YOU: TRIGONOMETRY
55. An airplane is traveling due south at 500 mph (the airplane vector is 𝐯𝐚 = −500 𝐣). The wind is blowing southeast at
40 mph (the wind vector is 𝐯𝐰 = 20√2𝐢 − 20√2 𝐣). The airplane’s actual vector as measured from the ground will be
given by the vector, 𝐯𝐠 = 𝐯𝐚 + 𝐯𝐰 . What is the airplane’s speed as measured from the ground? Round to the nearest
mile per hour.
(A) 473 mph
(B) 529 mph
(C) 645 mph
(D) 560 mph
(E) None of these
56. Solve the equation on 0 ≤ 𝑥 < 2𝜋: (sec 𝑥 + 2)(tan 𝑥 − 1) = 0.
2𝜋 4𝜋 𝜋 5𝜋
, , }
3 4 4
(B) { 3 ,
𝜋 4𝜋 𝜋 5𝜋
, , }
3 4 4
2𝜋 4𝜋 𝜋 7𝜋
, , , }
3 3 4 4
(E) None of these
(A) { 3 ,
(D) {
2𝜋 5𝜋 3𝜋 7𝜋
, , }
3 4 4
(C) { 3 ,
57. Solve the equation on 0 ≤ 𝑥 < 2𝜋: sin 𝑥 = sin 𝑥 cos 𝑥.
𝜋
(A) { }
2
58. Find the equivalent expression of
(A) sin 𝑥
𝜋 3𝜋
}
4 4
(B) {0, 𝜋}
cos2 𝑥
sin 𝑥
𝜋 3𝜋
}
2 2
(C) { ,
(D) { ,
(E) None of these
(C) sec 𝑥
(D) csc 𝑥
(E) tan 𝑥
(D) −120
(E) None of these
+ sin 𝑥.
(B) cos 𝑥
59. Let 𝐯 = −8𝐢 + 15𝐣 be a vector. Find the magnitude of 𝐯, ‖𝐯‖.
(A) 7
(B) 23
(C) 17
60. Find the force (independent of friction) required to keep a 1000-pound crate from sliding down a ramp that is inclined
10° to the horizontal. Round to the nearest pound.
(A) 100
(B) 177
(C) 985
(D) 174
(E) None of these
(B) 2√2
(C) √34
(D) 34
(E) 8
61. Find |𝑧| if 𝑧 = 5 − 3𝑖.
(A) 2
62. Compute the cube root of 8 and write your answer in the standard form 𝑥 + 𝑖𝑦.
(A) 2, −√3 ± 𝑖
(B) 2, 1 ± √3 𝑖
(D) 2, −1 ± √3 𝑖
(E) None of these
(C) 2, −1 ± 𝑖
63. Let v = −2𝐢 + 3𝐣 and w = 6𝐢 + 4𝐣. Find the dot product 𝐯 ∙ 𝐰.
(A) 0
(B) 11
(C) 24
(D) 10
(E) 90°
8
DR. YOU: TRIGONOMETRY
SOLUTIONS
1
B
16
D
31
C
46
D
61
C
2
C
17
E
32
C
47
C
62
D
3
D
18
A
33
A
48
A
63
A
4
D
19
C
34
B
49
B
5
C
20
E
35
B
50
C
6
A
21
D
36
B
51
C
7
C
22
B
37
C
52
C
8
C
23
E
38
E
53
B
9
E
24
B
39
C
54
A
10
B
25
B
40
A
55
B
11
B
26
B
41
B
56
A
12
D
27
D
42
B
57
B
13
D
28
B
43
D
58
D
14
B
29
A
44
E
59
C
15
A
30
D
45
B
60
D
9