math trigonometry
MATH 108 Trigonometry and Analytical Geometry Spring, 2018, V4.2
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MATH 108 FINAL EXAMINATION
This is an open-book exam. You may refer to your text and other course materials as you work on the exam, and you may use a calculator. You must complete the exam individually. Neither collaboration nor consultation with others is allowed. Record your answers and work on the separate answer sheet provided.
There are 23 problems. Problems #1–8 are Multiple Choice. Problems #9–16 are Short Answer. (work not required to be shown) Problems #17–23 are Short Answer with work required to be shown. MULTIPLE CHOICE
1. Use a calculator to find tan ���� � . Round the answer to 4 decimal places. 1. _______
A. −0.0439 B. −0.7265
C. 0.0439
D. 0.7265
2. For the triangle ABC, the following information is provided. B = 35°, a = 9, and b = 7. (Angle A is opposite side a, Angle B is opposite side b, Angle C is opposite side c.) State which case applies when solving the triangle and which law to use. 2. _______
A. Angle-Side-Side; law of sines
B. Angle-Side-Side; law of cosines
C. Side-Angle-Side; law of sines
D. Side-Angle-Side; law of cosines
3. Identify the type of conic section represented by the equation 2x 2 + 4y
2 – 8y – 9 = 0.
3. _______ A. parabola B. circle C. ellipse D. hyperbola
MATH 108 Trigonometry and Analytical Geometry Spring, 2018, V4.2
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4. Which of the functions corresponds to the graph? 4. ______
A. f (x) = sin x + 3
B. f (x) = cos(2x) + 2
C. f (x) = 3 – sin x
D. f (x) = 2 cos x + 1
5. The matrix below is an augmented matrix for a system of three linear equations. What can be concluded about the solution of the system? 5. _______
1 0 −50 1 00 0 0 � 2−11 �
A. The system is inconsistent; there is no solution.
B. There are infinitely many solutions. The solutions are (2 − 5t, −1, t) for all real numbers t.
C. There are infinitely many solutions. The solutions are (2 + 5t , −1, t) for all real numbers t.
D. The unique solution of the system is (2, −1, 1).
MATH 108 Trigonometry and Analytical Geometry Spring, 2018, V4.2
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6. Which of the equations corresponds to the graph? 6. ______
A. r = 1 + cosθ
B. r = 1 + sinθ
C. r = 1 + 2 cosθ
D. r = 1 + 2 sinθ 7. Consider the sequence 3, 7, 11, 15, 19, ... Is the sequence arithmetic or geometric, and what is the 40th term? 7. _______
A. geometric sequence; 159 B. geometric sequence; 163 C. arithmetic sequence; 159
D. arithmetic sequence; 163
8. Is the following the sum of an arithmetic sequence, a geometric sequence, or neither, and what is the sum?
� �95� ��
� � �
8. _______ A. sum of geometric sequence; sum = 21.3696 B. sum of geometric sequence; sum = 10.4976 C. sum of sequence which is neither arithmetic nor geometric; sum = 12.2976 D. sum of arithmetic sequence; sum = 21.3696
MATH 108 Trigonometry and Analytical Geometry Spring, 2018, V4.2
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SHORT ANSWER:
9. (a) Give an example of an angle which is coterminal with 480°. Answer: ________
(b) Convert the angle �� �� from radian measure into degree measure. Answer: ________
10. Find the exact values (no decimal approximations). (a) sec 570° Answer: ________
(b) tan (−120°) Answer: ________
11. Given y = 7 sin(6x – π), state the (a) period Answer: ________ (b) phase shift Answer: ________
12. Solve the trigonometric equation �cos ���√2 cos � − 1! = 0 in the interval [0, 360°). Find the exact values. Answer: _________________
13. (a) Find the exact value of arccos �tan ��� � Answer: ________
(b) Find the exact value of arcsin �sin ��% � Answer: ________
MATH 108 Trigonometry and Analytical Geometry Spring, 2018, V4.2
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14. (a) Given the point (6, 210°) in polar coordinates, find a corresponding point in rectangular coordinates. (State exact values, not decimal approximations.) Answer: _________________
(b) Given the point (–5, 5) in rectangular coordinates, find a corresponding point in
polar coordinates. Express the angle exactly in radians. (Your angle should include π). Answer: ________________
15. For the parabola given by (y + 7) 2 = 8(x – 4), find the following:
(a) direction parabola opens (to the left, right, up, or down) Answer: ___________ (b) vertex Answer: ___________ (c) focus Answer: ___________
16. Tickets at an amusement park cost $28 for children and $36 for adults. 1,500 tickets were sold, totaling $45,680 in receipts. Suppose you want to find out how many of each type of ticket were sold. Let x = number of children's tickets sold and y = number of adult tickets sold. (a) Write a system of two equations involving variables x and y, corresponding to the information given about the amusement park. Equation: ___________________________
Equation: ___________________________
(b) Find the solution of your system of equations from part (a). Solution: _________________
(c) Interpret the solution in a sentence, in terms of ticket sales at the amusement park.
Sentence: _____________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
MATH 108 Trigonometry and Analytical Geometry Spring, 2018, V4.2
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SHORT ANSWER, with work required to be shown, as indicated.
17. A building contractor is planning to use a ladder to reach a window. If an 20-ft ladder leans
against the building, with an angle of elevation of 72°, how far up the building will the ladder reach? Will the contractor be able to reach a window 18 feet above ground level? Show work. (sketch is not to scale)
18 . An ellipse has the equation �& ' (�)
%� + �+ , �� )
�% = 1 (a) Is the major axis horizontal or vertical? (b) Find the exact values of the foci of the ellipse. Show work.
19. Suppose that sin θ = 3/5 and that θ is a Quadrant II angle.
(a) Find the exact value of cosθ. Show work.
(b) Find the exact value of sin (2θ ). Show work.
20. Prove the identity 1 – (cos x – sin x) 2 = sin(2x)
72 o
20 ft
ladder
MATH 108 Trigonometry and Analytical Geometry Spring, 2018, V4.2
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21. For the triangle ABC, we are given that A = 40°, B = 63°, and c = 38.0.
Find the length of side b, rounded to the nearest hundredth. Show work.
22. Let -../ = 6, 4 and 0../ = 10, –15. (a) Calculate -../ + 10../. Show work.
(b) Calculate the exact length, or magnitude, of -../. Show work. (c) Calculate the dot product -../ ∙ 0../. Show work. (d) Determine the angle between -../ and 0../. Round the result to the nearest degree. Show work.
23. Solve the system of equations, using Gaussian elimination with or without matrices, or Gauss-Jordan elimination (your choice of method) Show work.
x – 2y + z = 3 2x + y + 2z = 11 –2x – y + 2z = 1