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What is the range of the function \(y=\log_2 (\sqrt{\cos x})\) for \(-90^\circ< x < 90^\circ\)?

 Aug 4, 2022
 #1
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\((-\infty,0]\)

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 Aug 4, 2022
 #2
avatar+124596 
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We only really need to look  at     0   ≤ x <   90

 

When  x  =  0

 

cos 0  = 1           sqrt ( cos 0)  = sqrt (1)  = 1

 

So y =    log2 1 =   0

 

When  x  is  almost  90

 

cos (almost 90)  =   almost 0 =  a very small positive

 

So

log  2   sqrt (a very small positive)   =   log 2   (a very small positive)  =  a very large negative

 

So...as the guest found, the range is  (-inf, 0 ]

 

cool cool cool

 Aug 4, 2022

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