If π/4 ≤ α < β ≤ π , which one is correct?

acemi123

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Apr 14, 2019
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If π/4 ≤ α < β ≤ π , which one is correct?

A. cos α > cos β
B. cos α < sin β
C. sin α > sin β
D. cos α < cos β
E. sin α < sin β

I guessed B, cos α < sin β but it is not. cosinus in first quadrant after π/4 must be lower than sinus. And in second quadrant cosinus is negatif there for i thought the answer was ''b'' but its not. Can anybody explain to me, please if I am wrong?
 

pka

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If π/4 ≤ α < β ≤ π , which one is correct?

A. cos α > cos β
B. cos α < sin β
C. sin α > sin β
D. cos α < cos β
E. sin α < sin β
Look at this plot. The graph is decreasing. What does say about A. ?
 

lev888

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Jan 16, 2018
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There is no guessing in math.
Look at this graph of sin and cos together. Can you answer the questions now?
 

JeffM

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Sep 14, 2012
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The reason B is wrong is this.

\(\displaystyle \alpha < \beta,\ \dfrac{ \pi }{4} \le \alpha < \pi, \text { and } \dfrac{ \pi }{4} < \beta \le \pi \implies\)

\(\displaystyle -\ 1 < cos( \alpha ) \le \dfrac{\sqrt{2}}{2} \text { and } 0 \le sin ( \beta ) < \dfrac{\sqrt{2}}{2}.\)

So, for example,

\(\displaystyle \alpha = \dfrac{\pi}{4} \text { and } \beta = \pi \implies \alpha < \beta,\ cos( \alpha) = \dfrac{\sqrt{2}}{2}, \text { and } sin ( \beta ) = 0 \implies\)

\(\displaystyle \alpha < \beta \text { and } cos ( \alpha ) >> sin ( \beta ).\)

It is a tricky problem because you may think that alpha and beta must be in the same quadrant, but nothing in the problem requires that. If alpha is in the first quadrant and beta is in the second, both the cosine of alpha and the sine of beta will be positive, and the cosine of alpha may exceed the sine of beta. It is of course true that if both were in the first quadrant or both were in the second quadrant, your reasoning would be correct, but the problem does not specify that.
 
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