Pre-Calculus Final Exam Review
Use the fundamental identities to simplify the expression.
1. cosx secx
4.
sec2 x  1
2
sin x
7. cosxsecx – cos2x
10.
1
1

1  cos x 1  cos x
2. sin2x (csc2x – 1)
3. cotx secx


 x  csc x
2

5.
cot x
csc x
6. sin 
8.
csc x cos x

sec x sin x
9. cot2x – cot 2 x cos2x
11. sin(-x)(1 tan2x)
12. Find the exact value of sine, cosine and tangent of 165˚.
13. Find the exact value given sinx = 3/5 and cosy = 12/13, and x and y are in quadrant I.
a) cos (x + y)
b) sin(x – y)
14. Solve the triangle given A = 37˚, a = 12 and B = 65˚.
15. Solve the triangle given A = 58˚, a = 4.5 and b = 12.8.
16. Find the area of the triangle with A = 86˚, b = 13 and c = 8.
17. From a certain distance, the angle of elevation to the top of a building is 17˚. At a point 50 meters
closer to the building, the angle of elevation is 31˚. approximate the height of the building.
18. Solve the triangle having sides of lengths 5, 7, and 10.
19. Solve the triangle with B = 12˚, a = 32 and c = 36.
20. To approximate the length of a marsh, a surveyor walks 425 meters from point A to B. Then the
surveyor turns 65˚ and walks 300 meters to point C. Approximate the length AC of the marsh.
21. Find the component form and magnitude of the vector with initial point (8, 5) and (-1, 4).
22. Use u = 6i – j and v = -3i + 5j to determine the following.
a) 3u – v
b) 2v + 5u
c) 2u
d) u ∙ v
23. Find a unit vector in the same direction of PQ . P(7, -4) and Q(-3, 2).
24. Find the angle between the vectors: 3, 5 and 2,8 .
25. Decide whether the vectors are parallel, orthogonal or neither.
a) 8, 4 and 5,10
b) 6,8 and 9, 12
26. Find the component form of the vector v with magnitude = 12 in the same direction as u = 3, 5 .
27. Forces with magnitude of 250 pounds and 130 pounds act on an object at angle of 45˚ and - 60˚,
respectively, with the x-axis. Find the direction and magnitude of the resultant of these forces.
28. Find the component form and magnitude of the vector with initial point (2, -2, 4) and terminal point
(-3, 2, -5).
29. Find the dot product of u and v. u = 3, 2,6 and v = 8, 2,1 .
30. Find a vector orthogonal to u and v. u = -3i + 2j – 5k and v = i + 2j + k.
31. Find the area of the parallelogram that has the vectors as adjacent sides.
u = i + 2j + 3k and v = 3i – k
32. Find the triple scalar product.
u = 1, 4,3 , v = 2,3, 5 , w = 3,0,6
33. Find the volume of the parallelepiped having adjacent sides u, v, and w.
u = 0, 2, 2 , v = 0,0, 2 , w = 3,0, 2
34. Find the equation of the plane passing through (2, 1, 3), (1, 3, -1) and (4, 0, 2).
35. Determine whether the planes are parallel, perpendicular or neither.
3x + 2y – z = 7 and x – 4y + 2z = 0
36. Expand the logarithms
a) log
5 y
x2
b) ln
x
x2
37. Condense to a single logarithm.
a) 4[ln z + ln (z + 5)] – 2ln (z-1)
b) log 4 – 3logx + ½ log y
Solve.
38. ex+2 = 8
39. log3(x + 5) – 8 = 12
41. ln x – ln 3 = 2
42. 4ln 3x = 15
40. -4(5x-3) = -28
43. The population of a town is modeled by P = 12,620e0.0118t , where t = 0 represents 2000. According to
this model, when will the population reach 17,000?
44. Write the first 5 terms of an 
45. Find the sum:
20
n
2
3n
.
n2
 3n
n 1
46. Write using sigma notation:
1 2 3
11
   ...  .
5 9 13
45
47. Write the first 5 terms of the arithmetic sequence with a4 = 12 and a16 = -72.
48. Write the first 5 terms given a1 = 3 and an+1 = 3an.
49. Find the sum (if possible).
a)
50
 5n  4
n 1
b)

2
3  

3
n 1
n 1
c)
12
 2  3
n 1
n 1
50. Find an for the following series: 27/2, 9, 6, 4, 8/3, 16/9, …
d)

3
2 

n 1  2 
n 1