FE_Review_F11_usethis (1)

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Unformatted text preview: MTH 142 Final Exam Formulas and Review Problems    Formulas  The following formulas will be provided for students on the final exam.  Students will  not be required to memorize these formulas.  All other definitions and formulas  given in the course must be memorized.  Students are not allowed to use their own  formulas sheets on the final exam.        Sum and Difference Identities:              Double Angle Identities:              Half Angle or Power Reduction Identities:                Area of an Oblique Triangle:                                    1 Problems  This review is simply a collection of problems from the semester to – it is not   comprehensive.  Problems 1 – 30 are multiple‐choice.  Problems 31 – 35 are free  response (show all work).    1.  Convert   to degrees:  A)   216     B)   108     C)   339.3     D)   1.88     E)  None of these      2.  The reference angle  for the angle   pictured below is given by    A)         B)          C)       D)     E)  None of these.                  θ 3.  Find the exact value of  A)     B)       :      C)   D)               E)  None of these  2 4.  Given    , where  .  If  equivalent to:    A)     B)    C)    D)    E)  None of these    5.  Convert  −165  to radian measure:  A)          is      B)    is the reference angle, then    C)       D)       E)  None of these    6.  The exact value of  A)    is:    B)       C)       D)  undefined   E)  None of these      7.  The reference angle for  A)    is:     B)       C)       D)       E)  None of these  3 8.  Match the graph below with the correct function:                                                   A)  y = 3 + csc x    B)  y = x + csc x     C)  y = 3 +  sec x  D)  y = x + sec x  E)  None of these    ⎛θ⎞ 9.   y = −2 sin ⎜ ⎟  has period:  ⎝ 3⎠ A)  2    B)  6      C)        D)         E)  None of these      10.  The range of  y = tan x  is:  ⎛ π π⎞ A)   ⎜ − , ⎟   ⎝ 2 2⎠ B)   ⎡ −1,1⎤   ⎣ ⎦ ⎧ π⎫ C)   ⎨ x | x ≠ odd integer multiples of ⎬  2⎭ ⎩ D)   −∞, ∞     ( ) E)  None of these    4 11.  What is the amplitude of  y = 3 − 4 cos( 2 x − 5π ) ?  A)  3      B)         C)  4      D)       E)  None of these    12.  The graph of  y = sec x  has vertical asymptotes at:  A)     B)     C)     D)  there are no vertical asymptotes    E)  None of these      13.  The area of the sector of a circle intercepted by a central angle of  60  in a circle  of radius 10 m is:    A)    sq.m     B)    sq.m     C)    sq.m      D)    sq.m      E)  None of these      14.  For the right triangle shown below, solve forside  x :  A)      B)     C)  14        D)         E)  None of these    5 15.    is:  A)       B)       C)       D)        E)  None of these      16.  An equivalent expression for  cot arctan x  is:  ( ) A)   x     1 B)       x 1 C)   2      x D)   x      E)  None of these      17.  The domain of  y = cos −1 ( x ) is  A)     B)     C)     D)    E)  None of these      18.  The range of  y = sin −1 ( x ) is  ⎡ π π⎤ A)   ⎢ − , ⎥   ⎣ 2 2⎦ B)   −∞, ∞   ( ) C)   D)      E)  None of these  6 19.  Simplify  :  A)      B)       C)       D)       E)  None of these    cos x 20.  Simplify  :  1 + tan 2 x A)   cos x + cos x cot 2 x    B)   cos x sin 2 x     C)   sec x     3 D)   cos x       E) None of these    21.  Use a sum identity to find the exact value of  sin 105 :  A)      B)      C)      D)      E)  None of these              22.  Find  :  A)      B)       C)         D)       E)  None of these  7 ⎛ π⎞ 23.  Use the difference identity to rewrite  cos ⎜ x − ⎟ :  4⎠ ⎝ A)   B)       2 cos x + sin x     2 ( ) C)   2 cos x sin x   D)  0      E)  None of these      24.  Find all solutions for the equation  sin x + 3 = − sin x   in the interval  A)        B)   C)   D)  :                       E)  None of these    25.  Find all solutions for the equation  cos( 2 x ) = A)   B)   2   in the interval  2 :            C)          D)           E)  None of these              8 26.  If  A)    and  , find  :     B)              C)       D)      E) None of these      27.  The radius of a circle is 8 m.  Find the length of an arc of the circle intercepted  by a central angle of  A)    m    B)   m    C)   m    D)    m  :    E)  None of these    28.  Given an oblique triangle with  C = 72 , A = 15 ,  and  b = 342.6 , find side  a .  A)  1258.92      B)  88.79     C)  6323.1     D)  326.28     E)  None of these      29.  Given an oblique triangle with  a = 80, b = 51,  and  c = 113 , find angle  C .  A)          B)        C)        D)        E)  None of these      9 30.  What is the angle of elevation (in degrees) to the top of a building 200 feet tall  from a point 12 feet from the base of the building?    A)   3.4        B)   93.4     C)   86.6      D)   12      E)  None of these    _________________________________________________________________________________________________      Show all work for Problems 31 – 35.    31.  Verify the identity:  sin 2 θ = sec θ − cosθ                  cosθ   32.  Solve the equation  2 cos 2 θ − cosθ = 1  for  θ  in  .      33.  To find the height of a building, a surveyor takes two measurements as  indicated in the figure below.  Use the information given below to find the height of  the building.  Round your answer to 2 decimal places                                        34.  A plane is flying in a race in which the course is in the shape of a triangle with  sides as given in the figure below.  Find the angle (in degrees) between the 220  miles and the 310 miles sides.  Round your answer to 2 decimal places.  Circle your  final answer.                                         10   ⎛ 3π ⎞ 35.   Verify the identity:   cos ⎜ + θ ⎟ = sin θ .  ⎝2 ⎠                                                                                              Answers        Multiple‐Choice  1. B  6. C  11. C  16. B  21. D  26. A  2. C  7. A  12. B  17. C  22. B  27. A  3. B  8. A  13. D  18. A  23. B  28. B  4. C  9. E  14. A  19. B  24. E  29. A  5. D  10. D  15. B  20. D  25. C  30. C          Free –Response       31.                          32.   sin 2 θ = sec θ − cosθ cosθ Work on RHS: 2 cos 2 θ − cosθ = 1 1 2 cos 2 θ − cosθ − 1 = 0 = − cosθ cosθ ( 2 cosθ + 1)(cosθ − 1) = 0 1 cosθ cosθ                         2 cosθ + 1 = 0 or cosθ − 1 = 0   = − cosθ cosθ 1 cosθ = − or cosθ = 1 1 − cos 2 θ 2 = cosθ 2π 4π θ = 0, , sin 2 θ 33 = cosθ               11     33.      Find the angles in the oblique triangle first, then use Law of Sines: sin 10o sin 15o = 200 x where x is the hypotenuse of the smaller right triangle.   Solving for x : x= o 200 sin 15 = 298.0958955 sin 10o Use right triangle trig on the smaller right triangle: h sin 25o = (h is the height of the building) x h = x sin 25o = ( 298.0958955) sin 25o = 125.98 ft     34.   Use Law of Cosines 1522 = 2202 + 3102 − 2( 220)(310) cos C 1522 − 2202 − 3102 = cos C −2( 220)(310)   .89 = cos C   35.   C = cos −1 (0.89) = 27.13o   ⎛ 3π ⎞ cos ⎜ + θ ⎟ = sin θ ⎝2 ⎠ Expand the left side using the sum identity: ⎛ 3π ⎞ 3π 3π cos ⎜ + θ ⎟ = cos cosθ − sin sin θ 2 2 ⎝2 ⎠      = 0 ⋅ cosθ − ( −1) ⋅ sin θ = 0 + sin θ = sin θ 12 ...
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## This note was uploaded on 04/05/2012 for the course MTH 142 taught by Professor John during the Fall '04 term at Moraine Valley Community College.

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