advanced math final exam review name: may...2 7. cos๐=โ 1 2 8. sec๐=undefined 9....
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Advanced Math Final Exam Review Name:__________________________ May โ June 2017 Use the following schedule to complete the final exam review.
Homework will be checked in every day.
Late work will NOT be accepted.
Homework answers will be provided at the beginning of each class period for you to
check your work from the previous night. FINAL EXAM SCHEDULE: The first exam each day begins promptly at 8:00 a.m. Friday, June 2nd 6th & 7th hour exams [90 min each]
Monday, June 5th 4th & 5th hour exams [90 min each]
Tuesday, June 6th 2nd & 3rd hour exams [90 min each]
Wednesday, June 7th 1st hour exam [90 min]
Day Date Assigned Assignment Page in Packet Thurs. May 18, 2017 Trig Part 1 #1-30 all Pg. 2-3
Tues. May 23, 2017 Trig Part 1 #31-37 all Trig Part 2 #1-13, odds, 14-16 all
Pg. 3 Pg. 4
Wed. May 24, 2017 Trig Part 2 #17-28 all, 30, 32 Pg. 5-6 Thurs. May 25, 2017 Trig Part 3 #1-11 all, 12, 13, 14, 15, 17, 18 Pg. 7-9 Fri. May 26, 2017 Exponential/Logs #1-19 odds Pg. 10-11 Tues. May 30, 2017 Exponential/Logs #2-18 evens Pg. 10-11 Wed. May 31, 2017 Vectors #2-20 evens, 21-23 all Pg. 12-14 Thurs. June 1, 2017 Matrices #5-20 all Pg. 15-16
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Trig Part 1 - SOH CAH TOA, Special Rt Triangles, & Unit Circle Fill in the ratios for each trig function using the letters for opposite (O), adjacent(A) and hypotenuse(H)
sin = cos = tan = csc = sec = cot =
Find each value. No decimals โ reduce fractions.
1. sin ๐ท 2. sin ๐ธ 3. cot ๐ธ 4. tan ๐ท
5. cos ๐ท 6. csc ๐ท 7. sec ๐ท 8. cot ๐ท
9. cos ๐ธ 10. tan ๐ธ 11. sec ๐ธ 12. csc ๐ธ
Using the 30-60-90 and 45-45-90 rules, find each length or angle measure. Express all lengths in simplest radical form.
13. AB = _____ BC = ______
A
6
45ยฐ
C B
14. PM = _____ MN = _____
P
15
30ยฐ
N M
15. AC = _____ AB = _____
A
60ยฐ
C 18 B
16. ST = _____ TV = ______
S
10
45ยฐ
T V
Fill in the missing sides of the special right triangles and answer the following questions. Exact answers only
17.
sin 30ยฐ = ________
cos 30ยฐ = ________
tan 30ยฐ = ________
18.
sin 60ยฐ = ________
cos 60ยฐ = ________
tan 60ยฐ = ________
19.
sin 45ยฐ = ________
cos 45ยฐ = ________
tan 45ยฐ = ________
20. Use a calculator to find each
value. Round to the nearest 100th.
sin 30ยฐ ______ cos 30ยฐ ______ tan 30ยฐ _______
sin 45ยฐ ______ cos 45ยฐ ______ tan 45ยฐ _______
sin 60ยฐ ______ cos 60ยฐ ______ tan 60ยฐ _______
Use SOH CAH TOA to solve for the following parts of the right triangles. C is a right angle.
21. c = 20, a = 15, find b
A
b c
C a B
22. a = 30, b = 12, find A
A
b c
C a B
23. A = 35, c = 9, find b
A
b c
C a B
24. a = 6, b = 12, find B
A
b c
C a B
F E
D
4
3
5
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Use SOH CAH TOA to solve each story problem
25. Katlyn leans a 16 foot ladder against the wall. If the ladder makes an
angle of 70ยฐ with the ground, how far up the wall does the ladder reach?
26. Tom leans a 20-ft ladder against a wall. The base of the ladder
is 4 feet from the wall. What angle ๐ does the ladder make with the
ground?
Find the reference angle for each.
27. 27ยฐ 28. 132ยฐ 29. 209ยฐ 30. 289ยฐ
31. Find the value of n to the nearest degree.
A. 53ยฐ B. 90ยฐ C. 45ยฐ D. 37ยฐ E. None of the above
32. A school flagpole casts a 16-foot shadow on the lawn. A teacher stood at the shadow's edge and measured the angle of elevation
to the top of the pole at 42ยฐ. How tall is the pole?
A. 12 ft B. 5.6 ft C. 17.8 ft D. 14.4 ft E. None of the above
33. Find the value of j rounded to the nearest tenth.
A. 663.4 B. 197.1 C. 49.3 D. 13.2 E. None of the above
34. Change 4๐
3 to degrees.
A. 45ยฐ B. 180ยฐ C. 250ยฐ D. 290ยฐ E. None of the above
35. Determine the exact value of sin 150ยฐ.
A. 1
2 B.
โ1
2 C.
โ3
2 D.
โโ3
2
E. None of the above
36. Find the exact value of sec2๐
3.
A. โ2โ3
3 B.
โ1
2 C.
2โ3
3
D. โ2 E. None of the above
37. Point ๐(0.8, 0.6) is located on a unit circle. Find sin ๐ and cos ๐.
A. sin ๐ = 0.6, cos ๐ = 0.8 B. sin ๐ = 0, cos ๐ = 1 C. sin ๐ = 0.8, cos ๐ = 0.6 D.sin ๐ = 1, cos ๐ = 0
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Trig Part 2 - More Unit Circle, Graphs, Identities, and Formulas Convert from degrees to radians Convert from radians to degrees
1. 45ยฐ 2. โ60ยฐ
3.
๐
5 4.
3๐
4
5. Complete all of the following.
90 Family
Sin 90
Cos 90
Tan 90
Sin 270 Cos 270 Tan 270
Csc 90 Sec 90
Cot 90
Csc 270 Sec 270 Cot 270
Sin 180
Cos 180 Tan 180
Sin 360 Cos 360 Tan 360
Csc 180
Sec 180
Cot 180 Csc 360 Sec 360 Cot 360
Give the exact value for each problem. 3600
6. sin ๐ = โโ3
2 7. cos ๐ = โ
1
2 8. sec ๐ = undefined
9. sin ๐ = โโ2
2
Draw a triangle for each problem and find the exact valueโ no decimals!!
10. Find sin ๐, if cot ๐ =3
4
and 0 < ๐ < 90ยฐ
11. Find sec ๐ if sin ๐ =1
3 and
90ยฐ < ๐ < 180ยฐ
12. Find cos ๐, if tan ๐ =8
15
and the angle is in the third
quadrant
13. Find cot ๐, if csc ๐ = โ5
and 270ยฐ < ๐ < 360ยฐ
Tell which quadrant each angle lies in. (I, II, III, or IV)
14. cos ๐ is positive and tan ๐ is negative.
15. csc ๐ is positive and tan ๐ is negative. 16. tan ๐ is positive and sec ๐ is negative.
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Give the amplitude, period and phase shifts (up/down/left/right) of each graph โ you do not need to graph these.
17. ๐ฆ = 2 sin(๐ โ 180ยฐ) + 1
Amplitude = Period =
h = _______ left or right Reflection? Y/N
k = ______ up or down
18. ๐ฆ = โ3 cos ๐ โ 4
Amplitude = Period =
h = _______ left or right Reflection? Y/N
k = ______ up or down
19. ๐ฆ =1
2sin(๐ + 90ยฐ)
Amplitude = Period =
h = _______ left or right Reflection? Y/N
k = ______ up or down
Directions: Answer the following questions for the given sine and cosine graphs. 20. Which equation is shown on the graph at the right:
A. ๐ฆ = cos ๐ โ 2
B. ๐ฆ = cos ๐ + 2
C. ๐ฆ = 2cos ๐
D. ๐ฆ =1
2cos ๐
21. Which equation is shown on the graph at the right:
A. ๐ฆ = โ sin ๐
B. ๐ฆ = sin ๐ + 1 C. ๐ฆ = sin ๐ โ 1
D. ๐ฆ = sin ๐ โ 2
22. What is the vertical shift shown on the graph at the right?
23. Which equation is shown on the graph at the right:
A. ๐ฆ = cos ๐ โ 2
B. ๐ฆ = cos ๐ + 2 C. ๐ฆ = 2cos ๐
D. ๐ฆ =1
2cos ๐
24. What is the amplitude shown on the graph at the right?
25. Which equation is shown on the graph at the right:
A. ๐ฆ = sin ๐ โ 2 B. ๐ฆ = sin ๐ + 2
C. ๐ฆ = 2sin ๐
D. ๐ฆ =1
2sin ๐
26. What is the amplitude shown on the graph at the right?
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Use your formula sheet to remind yourself of the trig identities.
Simplify. Show all work.
27. tan ๐ csc ๐
sec ๐ 28.
sin2๐โcot ๐ tan ๐
cot ๐ sin ๐
Use your formula sheet to remind yourself of the sum and difference formulas.
Use the sum and difference formulas to find the EXACT value of each trig. function. Rationalize when necessary!
29. cos 255ยฐ
30. sin 285ยฐ
31. tan 15ยฐ
32. cos 195ยฐ
33. sin 105ยฐ
34. tan 285ยฐ
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Trig Part 3 - Laws, Inverses, Solving, and Areas of Triangles
Law of Sines: sin ๐ด
๐ด=
sin ๐ต
๐=
sin ๐ถ
๐ Law of Cosines: ๐2 = ๐2 + ๐2 โ (2๐๐ cos ๐ด)
Determine whether to use the Law of Sines of the Cosines to solve the given triangles. Circle one. You do NOT need to solve!
1.
Sines
Cosines
2.
Sines
Cosines
Use the Law of Sines and/or the Law of Cosines to solve each triangle. Round all decimals to the nearest 100th
.
3.
โ ๐ด = ___________ โ ๐ต = _____________ ๐ = _______________
4.
โ ๐ด = ___________ โ ๐ต = _____________ โ ๐ถ = _______________
5. โ ๐ด = 32.2ยฐ, ๐ = 21.3, ๐ = 34.5
โ ๐ต = ___________ โ ๐ถ = _____________ ๐ = _______________
6. โ ๐ด = 127ยฐ, ๐ = 32, ๐ = 25
โ ๐ต = ___________ โ ๐ถ = _____________ ๐ = _______________
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7. An isosceles triangle has a base of 22 cm and a vertex angle of ๐๐ยฐ. Find the perimeter of the triangle.
8. In โ๐ด๐ต๐ถ, A = 32ยฐ, a =8, and b = 7. Find C.
A. 30.4ยฐ B. 110.7ยฐ C. 120.4ยฐ D. 20.7ยฐ E. None of the above
9. Find the length of side a to the nearest tenth.
A. 11.8 B. 8.7 C. 6.1 D. 10.9 E. None of the above
10. Solve sinโ1 โ3
2= ๐ (Recall restricted domain!)
A. 40.8ยฐ B. 56.3ยฐ C. 60ยฐ D. 30ยฐ E. None of the above
11. Solve cos ๐ = โโ2
2
A. 315ยฐ B. 225ยฐ C. 135ยฐ D. 45ยฐ E. None of the above
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Solve the following given that ๐ยฐ โค ๐ฝ โค ๐๐๐ยฐ. The number of solutions you should find is given in parentheses.
12. 2 tan ๐ sin ๐ โ 2 sin ๐ = 0 (5 solutions) 13. 2cos2 ๐ โ โ3 cos ๐ = 0 (4 solutions) 14. sin2 ๐ โ sin ๐ = 2 (1 solution)
The factored form is :
(sin ๐ โ 2)(sin ๐ + 1) = 0
15. 2 cos ๐ + 1 = 0 (2 solutions) 16. 2tan2๐ โ 5 tan ๐ + 3 = 0 (4 solutions)
The factored form is :
(2tan ๐ โ 3)(tan ๐ โ 1) = 0
17. 2cos2๐ = cos ๐ (4 solutions)
Use your formula sheet to find the area of the given triangle. Draw the triangle and then decide if you can find the area, or if you
have to use the Law of Sines or the Law of Cosines to find more information. Label your triangle as you find additional information. 18. ๐ด = 105ยฐ, ๐ = 12, ๐ = 24 19. ๐ต = 102ยฐ, ๐ถ = 27ยฐ, ๐ = 8.5 20. ๐ = 32, ๐ = 24, ๐ = 36
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Exponential Functions & Logs
Determine which of the following functions are exponential growth and which are exponential decay. Circle your answer. 1. ๐ฆ = 1.2๐ฅ GROWTH DECAY 2. ๐ฆ = 8(0.35)๐ฅ GROWTH DECAY
Graph the given exponential function using the following domain for : [โ๐, ๐]. State whether the graph shows exponential growth or decay.
3. ๐ฆ = 2๐ฅ 4. ๐ฆ =1
2
๐ฅ
Rewrite each equation in its exponential form.
5. log51
25= โ2 6. log 1000 = 3 7. log4 1024 = 5 8. ln ๐โ 6 = โ 6
Rewrite each equation in its logarithmic form.
9. 83 = 512 10. 121โ 1
2 = 1
11
Solve.
11. log๐ฅ 4 = 1
3 12. log6 216 = ๐ฅ 13. log3
1
81= ๐ฅ 14. ln ๐5 = ๐ฅ
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For the following application problems, show the equation (if one hasnโt already been given to you) and show the substitution of all the variables necessary to solve. 15. Kristin starts an experiment with 2,300 bacteria cells. The formula A = 2,300(1.14)t can be used to
approximate the number of bacteria cells after t hours. How many bacteria cells can be expected in the sample after 12 hours?
15.)______________ 16. An investment pays 3.8% interest compounded monthly. If $15,750 is invested in the account, what
will be the balance after 15 years?
16.)______________ 17. Youโre off to college! You buy a computer for $1400. The value of the computer can be modeled by
the equation V(t) = 1400(.85)t. What will be the value of the computer in 4 years?
17.)______________ 18. A college savings account pays 5.3% interest compounded continuously. What is the balance of an
account after 18 years if $20,000 was initially deposited?
18.)______________ 19. Freedom National Bank offers several types of savings accounts with different interest rates and
compounding periods.
Money-Market Account: Earns 2.8% interest, compounded quarterly. Standard Savings Account: Earns 3.35% interest compounded monthly. All-in-One Account: Earns 3.2% interest compounded continuously.
You have $2000 to deposit and plan to leave it in the account for 10 years. Which account is the best investment for your money?
Money Market Account Standard Savings Account All-in-One Account
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Vectors
Draw the vector from ๐ท๐ to ๐ท๐. Then find the vector in component form with the initial point ๐ท๐ and
terminal point ๐ท๐. Finally, draw the vector in component form.
1. ๐1 (โ2, 1) and ๐2 (5, โ6) 2. ๐1 (6, โ1) and ๐2 (4, 3) 1.)_______________
2.)_______________
Find the magnitude and direction for the given vector. Draw the vector to help decide the direction!
Estimate the magnitude to the nearest tenth. Estimate the direction to the nearest tenth of a degree.
3. ๐ = โจ5, โ12โฉ 4. ๐ = โจโ4, โ7โฉ 3.)_______________ Magnitude
3.)_______________ Direction
4.)_______________ Magnitude
4.)_______________ Direction
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Given ๐ = โจโ๐, ๐โฉ, ๐ = โจโ๐, โ๐โฉ and ๐ = โจ๐, ๐โฉ. Find the following.
5. โ๐โ 6. โ๐โ 7. โ๐ + ๐โ 5.)_______________
6.)_______________
7.)_______________
8. โ๐ + ๐โ 9. โ5๐ 10. 2๐ โ 3๐ 8.)_______________
9.)_______________
10.)______________
11. 1
2๐ + 5๐ 12. โ3๐ โ 4๐โ 11.)______________
12.)______________
Find the unit vector in the direction ๐ of by dividing each component of ๐ by the magnitude of ๐.
13. ๐ = โจ8, โ6โฉ 14. ๐ = โจ15, โ8โฉ 13.)______________
14.)______________
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Given ๐ = ๐๐ โ ๐ and ๐ = โ๐๐ + ๐๐, find the following:
15. 2๐ โ 3๐ 16. โ5๐ 17. โ๐โ 15.)______________
16.)______________
17.)______________
18. ๐ + ๐๐ 19. โ๐ + ๐๐โ 20. 1
2๐ โ
1
2๐ 18. )______________
19.)______________
20.)______________
Draw vectors ๐ and ๐ on the same coordinate plane. Then, find the dot product of ๐ and ๐, โ๐โ and
โ๐โ. Finally, find the measure of the angle between the two vectors. Then determine if the vectors are
orthogonal.
21. ๐ = โจ4, โ9โฉ ; ๐ = โจ2, 6โฉ 22. ๐ = โ3๐ + 2๐ ; ๐ = โ6๐ โ 9๐ 23. ๐ = โจ1, 2โฉ; ๐ = โจโ3, 7โฉ
๐ โ ๐ = ๐ โ ๐ = ๐ โ ๐ =
โ๐โ = โ๐โ = โ๐โ =
โ๐โ = โ๐โ = โ๐โ =
Angle between Angle between Angle between
two vectors = ___________ two vectors = ___________ two vectors = ___________
Orthogonal? Yes No Orthogonal? Yes No Orthogonal? Yes No
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Systems and Matrices
Solve each system by hand. State which method you used: Substitution or Elimination.
1. ๐ฆ = 40๐ฅ + 20
๐ฆ = โ10๐ฅ โ 30
(_______, _______)
Method = _______________
2. 6๐ฅ + 3๐ฆ = 6
8๐ฅ + 5๐ฆ = 12
(_______, _______)
Method = _______________
3. 2๐ฅ โ 3๐ฆ = โ2
2๐ฅ + ๐ฆ = 24
(_______, _______)
Method = _______________
4. 4๐ฅ โ 10 = ๐ฆ
2๐ฅ = 12 โ 3๐ฆ
(_______, _______)
Method = _______________
State the dimensions of each matrix.
5. [5 00 55 0
] 6. [
2 โ5 31 โ4 6
] 7. [
123
] 8. [4 5 โ6]
For #9-#10, use the matrices below.
๐ = [3 โ5
โ2 โ2] ๐ = [
โ1 94 2
]
For #11-#12, use the matrices below.
๐ = [1 โ5
โ2 2] ๐ = [
0 4 โ36 1 5
]
9. ๐ + ๐
10. 2๐ โ 3๐ 11. ๐ถ๐ท 12. ๐ท๐ถ
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Use the following matrices for the multiplication problems:
๐จ = [1 โ2
12 10] ๐ฉ = [
โ13 615 โ1
] ๐ฎ = [0 11 โ126 1 5
โ3 4 5] ๐ฏ = [
โ2 โ5 10โ4 0 122 โ33 11
]
13. ๐ด๐ต 14. ๐ต๐ด 15. ๐บ๐ป
16. What are the dimensions
of the product matrix when the
following two matrices are
multiplied: 5๐ฅ2 โ 2๐ฅ3?
17. Find the inverse of matrix ๐ฝ = [1 24 3
] using your calculator. 18. Find the inverse of matrix ๐พ = [โ2 35 โ1
] using your
calculator.
Use the inverse matrix and matrix multiplication to solve the following systems. Start by filling in the Coefficient, Variable and
Constant matrices. Remember to write your answer as an ordered pair or ordered triple.
19. 3๐ฅ โ 2๐ฆ = 7
5๐ฅ + 3๐ฆ = โ1
AX=B
[ ] [ ] = [ ]
๐ดโ1๐ต = [ ]
(_______, ________)
20. 2๐ฅ โ ๐ฆ + 2๐ง = 15
โ๐ฅ + ๐ฆ + ๐ง = 3
3๐ฅ โ ๐ฆ + 2๐ง = 18
AX=B
[ ] [ ] = [ ]
๐ดโ1๐ต = [ ]
(_______, ________, _______)