[answered] MATH 108 Trigonometry and Analytical Geometry Fall, 2016 4.

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MATH 108 Trigonometry and Analytical Geometry Fall, 2016 4.1 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.

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

A. ?0.3827 B. 0.3827 C. ?0.9960 D. 0.9960 cos ???

? . Round the answer to 4 decimal places. 1. _______ 2. For the triangle ABC, the following information is provided. A = 65?, b = 9, and c = 16.

(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. Side-Angle-Side; law of cosines B. Side-Angle-Side; law of sines C. Angle-Side-Side; law of cosines D. Angle-Side-Side; law of sines

2 2 3. Identify the type of conic section represented by the equation 2x ? 2y ? 12y ? 3 = 0.

3. _______

A.

B.

C.

D. parabola

circle

ellipse

hyperbola Page 1 of 7 MATH 108 Trigonometry and Analytical Geometry Fall, 2016 4.1 4. Which of the functions corresponds to the graph? 4. ______ A. f (x) = sin(2x) + 1

B. f (x) = 2 sin x + 1

C. f (x) = 2 cos x + 1

D. f (x) = 2 ? cos x 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. _______ 4

1 0 0

0 1 3 ? 6 0 0 0

0

A. The unique solution of the system is (4, ?6, 0).

B. There are infinitely many solutions. The solutions are (4, ?6 + 3t, t) for all real numbers t.

C. There are infinitely many solutions. The solutions are (4, ?6 ? 3t, t) for all real numbers t.

D. The system is inconsistent; there is no solution. Page 2 of 7 MATH 108 Trigonometry and Analytical Geometry Fall, 2016 4.1 6. Which of the equations corresponds to the graph? 6. ______ A. r = 1 + cos?

B. r = 1 + sin?

C. r = 1 + 2 cos?

D. r = 1 + 2 sin?

7. Is the following the sum of an arithmetic sequence, a geometric sequence, or neither, and what

is the sum?

?

? 2

?? ?

5 ??? A.

B.

C.

D. sum of arithmetic sequence; sum = 0.6496

sum of geometric sequence; sum = 0.6496

sum of geometric sequence; sum = 0.4256

sum of sequence which is neither arithmetic nor geometric; sum = 0.0256 8. Consider the sequence 3, 10, 17, 24, 31, ...

Is the sequence arithmetic or geometric, and what is the 30th term?

A.

B.

C.

D. 7. _______ 8. _______ arithmetic sequence; 213

arithmetic sequence; 206

geometric sequence; 213

geometric sequence; 206

Page 3 of 7 MATH 108 Trigonometry and Analytical Geometry Fall, 2016 4.1 SHORT ANSWER: 9. (a) Give an example of an angle which is coterminal with 585?.

(b) Convert the angle 10. ??

?? Answer: ________ from radian measure into degree measure. Answer: ________ Find the exact values (no decimal approximations).

(a) sec 675? Answer: ________ (b) tan (?150?) Answer: ________ 11. Given y = 7 sin(4x ? ?), state the

(a) period Answer: ________ (b) phase shift Answer: ________ 12. Solve the trigonometric equation

Find the exact values. (cos x)(2sin x ? 1) = 0 = 0 in the interval [0, 360?).

Answer: _________________ 13. Find the exact value of arcsin ?sin ?? (b) Find the exact value of arccos ?sin ?? (a) ? ? ? ? Answer: ________

Answer: ________ Page 4 of 7 MATH 108 Trigonometry and Analytical Geometry Fall, 2016 4.1 14. (a) Given the point (4, 330?) in polar coordinates, find a corresponding point in

rectangular coordinates. (State exact values, not decimal approximations.)

(b) Given the point (? 8, 8) in rectangular coordinates, find a corresponding point in

polar coordinates. Express the angle exactly in radians. (Your angle should include ?).

Answer: ________________ 2 15. For the parabola given by (y + 5) = 8(x ? 3), find the following:

(a) direction parabola opens (to the left, right, up, or down) Answer: ___________ (b) vertex Answer: ___________ (c) focus Answer: ___________ 16. Tickets for a show were available at a discounted price of \$15 if bought in advance, and at

the full price of \$24 if purchased at the door. \$13,335 was collected through the sale of 715

tickets. Suppose you want to find out how many of each type of ticket were sold.

Let x = number of discount tickets sold and y = number of full-price tickets sold

(a) Write a system of two equations involving variables x and y, corresponding to the information

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 for the show.

Sentence: _____________________________________________________________

_________________________________________________________________________ Page 5 of 7 MATH 108 Trigonometry and Analytical Geometry Fall, 2016 4.1 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 70?, 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) 20 ft

ladder o 70 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 cos ? = 4/5 and that ? is a Quadrant IV angle.

(a) Find the exact value of sin?. Show work. (b) Find the exact value of sin (2? ). Show work. 20. Prove the identity (sin x + cos x)2 ? 1 = sin(2x) Page 6 of 7 MATH 108 Trigonometry and Analytical Geometry Fall, 2016 4.1 21. For the triangle ABC, we are given that A = 44?, B = 62?, and c = 30.0. Find the length of side a, rounded to the nearest tenth. Show work. ?? = ?8, ? 4? and ?

?? = ?6, 12?.

22. Let ?

(a) ?? + ??

??. Show work.

Calculate ? (b) ??. Show work.

Calculate the exact length, or magnitude, of ? (c) ?? ? ?

??. Show work.

Calculate the dot product ? (d) ?? and ?

??. Round the result to the nearest degree. Show work.

Determine the angle between ? 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 = 3

? x ? 3y + 2z = ?2 Page 7 of 7

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