winter'2015'math'112'final'exam''' ' ' name '...
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Winter'2015'Math'112'Final'Exam''' ' ' Name_______________________________'''Instructions:'Show'ALL'work.'Simplify'wherever'possible.'Clearly'indicate'your'final'answer.'''
Problem'Number' Points'Possible' Score'
1' 25' '
2' 25' '
3' 25' '
4' 25' '
5' 25' '
6' 25' '
Subtotal' 150' '
Extra'credit' 10' '
Total' 150' '
'' '
____________________________________________________________________________________________''
sin A + B( ) = sinAcosB + cosAsinB ''sin A − B( ) = sinAcosB − cosAsinB 'cos A + B( ) = cosAcosB − sinAsinB 'cos A − B( ) = cosAcosB + sinAsinB '
tan A + B( ) = tanA + tanB1− tanA tanB
''
tan A − B( ) = tanA − tanB1+ tanA tanB
'
'sin 2x( ) = 2sin xcos x ''cos 2x( ) = cos2 x − sin2 x '' ''= 1− 2sin2 x '
''= 2cos2 x −1 ''
tan 2x( ) = 2 tan x1− tan2 x
'
sin x2
⎛⎝⎜
⎞⎠⎟ = ± 1− cos x
2''
cos x2
⎛⎝⎜
⎞⎠⎟ = ± 1+ cos x
2'
tan x2
⎛⎝⎜
⎞⎠⎟ =
1− cos xsin x
'
'= sin x1− cos x
'1) Indicate'whether'each'of'the'following'statements'are'true'or'false'(no'explanation'necessary):''
a) If' sin x = 35'then' cos x 'must'equal' 4
5.'
'''
b) arcsin sin 3π4
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟= 3π4''
'''
c) arccos cos 3π4
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟= 3π4'
''''
d) If'270 <α < 360 ,'then' cosα 'must'be'negative''
''''
e) If'270 < 2α < 360 ,'then' tanα 'must'be'negative'''''
f) tan495! = −1 .'''''
g) 2 − 2cos2 x = 2sin2 x ''''''
h) sec 7π6
= − 2 33
'
''''
i) Brian'will'finally'win'the'March'Madness'office'pool'this'year.'' '
2) Solve'the'equations'below'for'x.#Simplify'your'answer'so'that'it'does'not'involve'logs.''
a) 2 log4 x − log4 2 = 8 'Hint:'Only'the'positive'solution'works''''''''''''''
b) log2 x + 3( ) + log2 x + 2( ) = 3− log3 9 ''Hint:'The'right'side'of'the'equation'is'a'whole'number.''''''''''''''
''
c) 32 ⋅2x−1 = 8x+1 'Hint:'Answer'is'a'fraction.'
'' '
3) Trig'Equations.'Find'all'solutions'that'lie'in'the'interval' 0, 2π[ ) ,'to'the'equations'below.''
a) sin3 x
cos x− tan x = 0 ''
''''''''''''''
b) 2sin x − cos2 x = 4 ''''''''''''''''''
c) 8sin2 xcos x + 4sin2 x − 6cos x = 3 '''
' '
4) Simplify'the'following'so'that'they'do'not'involve'any'trig'functions.''Hint:'Draw'triangles''
a) sin arccos 53
⎛⎝⎜
⎞⎠⎟
⎛
⎝⎜⎞
⎠⎟'
''''''''''''''
'b) sin arccos x( ) ''
''''''''''''''
'
c) tan arcsin 610
⎛⎝⎜
⎞⎠⎟ + arccos
1213
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟'
'' '
5) Word'problems'made'from'the'first'two'pictures'that'google'images'provided'me.''
a) How'far'is'the'ship'below'from'the'port?'
Assume' cos 160!( ) = −136
160'(it'doesn’t,'but'I'want'you'to'simplify'your'answer).''
'
''''''b) In'the'picture'below'(not'drawn'to'scale),'the'distance'from'B'to'the'top'of'the'tower'is' 24m ,'and'
the'angles'shown'below'measure' α = 60! 'and' β = 15! .'How'far'apart'are'A'and'B?'Simplify'your'answer.'
'
''
6) Simplify'the'following:' ' ' ' ' ' ' ' ' ' ''
a) sin x + cos x( ) sin x − cos x( ) + sin x + cos xsin x + cos x
''
''''''''''''''''''''
'
b) sin2 x + 2sin2 x + sin x − 32sin x + 3
+ cos2 x ''
' '
EXTRA'CREDIT:'Briefly'explain'why'each'line'is'equal'to'the'line'above'it.''1− 4 cos2 log y( )cos log x( )sin log x( ) + cos2 log x( )cos log y( )sin log y( )⎡⎣ '
−sin2 log x( )cos log y( )sin log y( )− sin2 log y( )cos log x( )sin log x( )⎤⎦2 '
'''
1− 2⎛⎝ cos2 log y( )cos log x( )sin log x( ) + cos2 log x( )cos log y( )sin log y( )⎡⎣ ''
−sin2 log x( )cos log y( )sin log y( )− sin2 log y( )cos log x( )sin log x( )⎤⎦⎞⎠⎟2
'
'''
= 1− 2 sin log x( )cos log y( ) + cos log x( )sin log y( )( ) cos log x( )cos log y( )− sin log x( )sin log y( )( )⎛⎝
⎞⎠
2
'
''''
= 1− 2sin log x + log y( )cos log x + log y( )⎛⎝
⎞⎠
2
'
''''
= 1− 2sin log xy( )( )cos log xy( )( )⎛⎝
⎞⎠
2
'
''''= 1− sin2 2 log xy( )( ) '''''= 1− sin2 log xy( )2( ) ''''''= cos2 log xy( )2( ) ''''
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