Help with math final exam homework

MATH 108 Fall 2016 Final Exam

Problems 1 to 3. 2 pts. each. Find the exact values, with no decimal approximations.

1. cos (585°)

2. tan(-5π/3)

3. csc(11π/6)

4. 7 pts. Find the exact values, with no decimal approximations.

Given that cotθ = 9 for an angle θ between π and 3π/2,

A. Find cosθ.

B. Find sin(2θ).

C. Find cot(π/2 - θ)

5. 2 pts. Use a calculator to find the function value to four decimals: tan(7π/15)

6. 3 pts. Use a calculator to find θ, such that sec θ = -4.8097 and θ is between 180° and 270°.

7. 3 pts. Solve. A small airplane is flying at an altitude of 3480 ft. The angle of elevation from the airport

to the plane is 37°24’10”. How far is the plane from the airport? Answer should be accurate to the nearest

tenth of a foot.

8. 3 pts. Rewrite this sum using sigma summation notation. -312 - 299 - 286 - 273 - 260 - 247 - 234

9. 3 pts. A bike tire with a diameter of 26 inches rolls forward 1 foot. Through what angle (to the nearest

tenth of a degree) did the wheel rotate?

10. 2 pts. Find arccot(0)

11. 2 pts. Find arcsin(cos4π/3)

12. 3 pts. Convert the point given in rectangular coordinates to polar coordinates: (6, -2√3). Find exact

values, with no decimal approximations.

13. 4 pts. Vector u is <5, -16> and vector v is <-8, 12>.

A. Find the magnitude of vector u.

B. Find the dot product u ∙ v

C. Find the angle between vectors u and v to the nearest tenth of a degree.

14. 3 pts. Evaluate the sum ∑ (−1)𝑘 ∙ ( 2𝑘+1

𝑘 )4𝑘=1

15. 4 pts. A parabolic shaped tunnel is 12 feet wide at the base and 15 feet tall. Find an equation of a

parabola that would model this tunnel if the vertex is placed at (0,0).

16. 5 pts. A. Graph the polar equation r = 1 + 2sinθ. (You can use a graphing calculator.)

Complete the polar coordinates in the form (r, theta) and quadrants for these two polar points which

satisfy this equation.

B. (____, 5π/4). This polar point falls in quadrant ____.

C. (____, -7π/6). This polar point falls in quadrant ____.

17. 5 pts. Leaving home, a man drives south for 72 miles, and then turns and goes S36°E for 110 miles.

How far is the man from home (to the nearest tenth of a mile)? What direction is he from home? (That is,

find the bearing from home to the man’s current location.)

18. 8 pts. For the function y =2sin[x/4 - π/2]

A. Graph at least one period of the function, and find the amplitude, period, and phase shift.

B. Find the y coordinate for this point (4π/3, ___) which falls on the graph of the given function.

C. Find an x coordinate for a point (___, 0) which falls on the graph of the given function.

19. 4 pts. A. Find the explicit formula, or general rule an, for the sequence: 270, 90, 30, 10, …..

B. Find the sum S∞ of all the terms in the sequence: 270 + 90 +30 + 10 +….

20. 5 pts. Graph the equation. Find the vertices, foci, and center point.

25x2 + 4y2 + 50x + 4y + 1 = 0

21. 4 pts. Find the equation of the hyperbola satisfying the given conditions.

Vertices at (0, -8) and (0,4); asymptotes with slopes ±1/3.

22. 6 pts. Solve the system of equations using only the method of Gaussian elimination on an augmented

matrix. Use row-equivalent operations on the matrix until it is in row-reduced echelon form and looks like

the identity matrix in the first three columns. You must show all work to receive credit.

2x - 3y - 4z = -9

4x + y + 3z = -14

-6x + 5y + 9z = 5

23. 5 pts. Prove the identity. Use algebra and basic trigonometric formulas to transform one side of the

equation to look like the other side. Show the whole process, without skipping any steps, and explain

what you are doing in each step.

𝑡𝑎𝑛𝑥 − 𝑠𝑖𝑛𝑥

2𝑡𝑎𝑛𝑥 = 𝑠𝑖𝑛2(

𝑥

2 )

24. 4 pts. Solve, finding all solutions in [0, 2π): 2sin2(x + π/6) = 1

25. 4 pts. Solve for all x: 3tan3 x - tanx = 0

26. 5 pts. Solve the triangle(s). Angle A is opposite side a, angle B is opposite side b, and angle C is

opposite side c. Round your answers to the nearest tenth.

a=16, b=12, B = 42°.