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' '

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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( ) ''''