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[Solved] Trigonometry using the roots

 
Roots
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June 14, 2021 6:13 pm
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(@armor)
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June 14, 2021 6:13 pm
Topic starter
Staff 
(@armor)
Admin
  Honorable Member
317 Posts
156 153 0

Assume the two roots are \small x_{1},x_{2}>0.  Since \small \Delta =(-4)^2-4(2cos\Theta -1)(4cos\Theta +2)>0,  then  \small -\frac{\sqrt{3}}{2}<cos\Theta <\frac{\sqrt{3}}{2}       (1)

Since \small x_{1}+x_{2}=\frac{4}{2cos\Theta -1}>0,   then      \small cos\Theta >\frac{1}{2}       (2)

Since   \small x_{1}x_{2}=\frac{4cos\Theta +2}{2cos\Theta -1}>0,   then    \small cos\Theta <-\frac{1}{2}    or     \small cos\Theta >\frac{1}{2}       (3)

According to (1), (2), (3), we can obtain \small \frac{1}{2}<cos\Theta <\frac{\sqrt{3}}{2}.   Sİnce  \small \Theta  is an acute angle, then \small 30^{\circ}<\Theta <60^{\circ}.

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