Download AdvPreCal Spring Final Exam Review 1. Given sin x = 4/7 and cos x

AdvPreCal Spring Final Exam Review
1. Given sin x = 4/7 and cos x < 0, find cot x.
2. Simplify:
csc 𝑥 𝑐𝑜𝑠 2 𝑥
1+csc 𝑥
tan 𝑥
3. Simplify:
csc 𝑥
+
sin 𝑥
tan 𝑥
4. Simplify:
tan 𝑥
1−sec 𝑥
For 5-7, find all solutions in the interval 0, 2  :
6. sec2x – 3 tan x = 5
5. csc x + 2 = 0
8. Evaluate: sin 255⁰
9. Evaluate: 𝑡𝑎𝑛
7𝜋
12
(Use the fact that 105 =210⁰+45⁰)
(Use the fact that
10. Find the exact value:
7. 2 sin x cos x + cos x = 0
7𝜋
12
tan 325°−𝑡𝑎𝑛25°
1+𝑡𝑎𝑛325°𝑡𝑎𝑛25°
=
3𝜋
4
−
𝜋
6
.
11. Simplify: cos146⁰cos11⁰+sin146⁰sin11⁰
12. Given cot u = 2/5 , 0 < u < 𝝿/2, and cos v = -3/5, 𝝿 < v < 3𝝿/2, find tan(u + v).
13. Given: sin x = -1/8 and tan x < 0, find sin 2x.
14. Given triangle with B = 87⁰, C = 24⁰, and a = 113, find b.
15. Given triangle with a = 18, b = 23, and B = 97⁰, find C.
16. Given triangle with a = 72, b = 51, and A = 27⁰, find the area of the triangle.
17. A hot air balloon is tethered to the ground with two 76 foot ropes. The angle of elevation
each makes with the ground is 24⁰. How far apart are the two ropes?
18. Given a triangle with a= 7, b=5, and c = 6, find A.
19. Given triangle with A = 38⁰, b = 22, and c = 98, find a.
20. Use Heron’s formula to find the area of a triangle with a = 42, b = 51, and c = 57.
21. A boat leaves port and sails 16 miles at a bearing of S 20⁰ E. Another boat leaves the
same port and sails 12 miles at a bearing of S 60⁰ W. How far apart are the two boats?
22. A vector v has initial point (3, 7) and terminal point (3, -2). Find its component form.
23. Determine the magnitude of the vector v: v = 7i – 2j.
24. Find the direction of v:
v = 〈3, −7〉
25. Given v of magnitude 50 and direction 315⁰, and w of magnitude 20 and direction 206 ,
find v + w.
26. Given w = 2i – 3j and v  2i  5j , find w v .
27. Rewrite in trigonometric form: -2 + 3i
28. Rewrite in standard form: 5(cos 120⁰ + isin120⁰)
29. Multiply: [3(cos85⁰ + isin85⁰)][12(cos10⁰+isin10⁰)]
30. Evaluate: (3 +3i)8.
31. Find the number of distinguishable ways the letters OKEECHOBEE can be arranged.
32. Determine the number of seven digit telephone numbers that can be made under the
condition that each of the first three digits cannot be zero.
33. An auto license plate is made using three letters followed by two digits. How many
license plates are possible?
34. How many ways can 4 girls be selected from a group of 30 girls?
35. A box holds 11 white, 4 red, and 8 black marbles. If 2 marbles are picked at random,
without replacement, what is the probability that they will both be black?
36. Find a formula for the nth term of the sequence.
3 4 5
6
{1 , 2 , 6 , 24 , … }
37. Find the sum:
38. Find the 54th term of the sequence 10, 6, 2, -2, …
39. Find the 24th term of the geometric sequence with a1 = 3 and r = 1.2.
40. Find the sum of the first 30 terms in the geometric sequence: 4, 26/5, 169/25, …
41. Evaluate:
42. Evaluate: lim𝑛→∞
lim𝑛→∞ (1 +
2𝑛−8
𝑛3 +11
2𝑛4 −5𝑛
3𝑛4 +7𝑛−3
43. Evaluate: lim𝑛→−3
𝑥 3 +27
𝑥+3
)
45. Evaluate using the first 4 terms of the power series: cos
5𝜋
𝑥 2 +3𝑥−88
(𝑥+2)(𝑥−3)2
48. Using the function 𝑓(𝑥) =
find the y-intercept(s).
𝑥 2 +3𝑥−88
find the x-intercept(s).
(𝑥+2)(𝑥−3)2
49. Find the vertical asymptote(s) for f(x) =
50. Find the horizontal asymptote for f(x) =
51. Find the slant asymptote for f(x) =
9
𝑒6
46. Evaluate using Euler’s formula:
47. Using the function f(x) =
𝜋
𝑥−5
.
𝑥 2 −6𝑥+5
3𝑥 2 +8𝑥−5
4𝑥 2 +𝑥−6
.
𝑥 2 +5𝑥+4
𝑥−3
52. Identify the removable discontinuity for 𝑓(𝑥) =
2𝑥 2 +5𝑥+3
𝑥 2 +𝑥
44. Evaluate:
53. Find the partial fraction decomposition of
𝑥−16
.
𝑥 2 −2𝑥−8
54. Find three other ways to represent the polar coordinate (3, 2𝝿/3)
55. Convert from rectangular to polar coordinates: (−√3, −1)
56. Change from polar to a rectangular equation: rsin𝚹 = -4.
57. Change from rectangular to a polar equation: x2 + y2 = 25
58. Graph each equation:
a. r = 5 – 3sin 𝚹 b. 𝑟 2 = 16𝑐𝑜𝑠2𝜃 c. r = 2sin3𝚹
d. r = 5 + 4sin𝚹
59. Find an equation for the graph at the right.
a. r  3sin 4
b. r  3cos 2
c. r  3  4cos
d. r = 5cos2𝚹
60. Eliminate the parameters given x = 3t - 1 and y = t/4.
61. Write y = x2 – 4 in parametric form with the parameter t = x – 1.
62. Eliminate the parameters
x = 4cos t and y = 5sin t.
63. Graph this parametric equation.
T [-π, π] Tstep 0.1
x1 = 10cos(t) + 4cos(9t)
y1 = 10sin(t) + 4sin(9t)
X [-30, 30] Xscl 5
Y [--20, 20] Yscl 5