anonymous
  • anonymous
sin²θ-cos²θ-cosθ=1. Find all solutions in the interval [0,2π).
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
\[sin^2(x)-cos^2(x)-cos(x)=1\] first rewrite in term of cosine alone, using the identity \[sin^2(x)+cos^2(x)=1\] \[1-cos^2(x)-cos^2(x)-cos(x)=1\] \[1-2cos^2(x)-cos(x)=1\] \[-2cos^2(x)-cos(x)=0\] \[2cos^(x)+cos(x)=0\] \[cos(x)(2cos(x)+1)=\] \[cos(x) = 0\] or \[2cos(x)+1=0\]
anonymous
  • anonymous
of course you still have to solve \[cos(x) = 0\] \[x=\frac{\pi}{2}\]or \[x=\frac{3\pi}{2}\]
anonymous
  • anonymous
\[2cos(x)+1=0\] \[cos(x)=-\frac{1}{2}\] \[x=\frac{2\pi}{3}\] \[x=\frac{4\pi}{3}\]

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anonymous
  • anonymous
so, x=2π/3 and 4π/3?
anonymous
  • anonymous
you have 4 solutions all together.
anonymous
  • anonymous
oh, ok...also the π/2 and 3π/2.... thanks
anonymous
  • anonymous
right. is it clear?
anonymous
  • anonymous
it is, when someone shows you!
anonymous
  • anonymous
mmm rly u have unfintly solutin here by adding the period of ur functin
anonymous
  • anonymous
of course! idea is to write using one trig function, then you get a quadratic equation to solve. there are an infinite number of solutions, that is why the question restricts interval from \[[0,2\pi)\]
anonymous
  • anonymous
i.e ur sol + 2n bay
anonymous
  • anonymous
thanks 'satellite 73' . hard to know where to start.
anonymous
  • anonymous
welcome!
anonymous
  • anonymous
mm sry 'm jst dont care 2 the question
anonymous
  • anonymous
it is, when someone shows you!
anonymous
  • anonymous
oh, ok...also the π/2 and 3π/2.... thanks
anonymous
  • anonymous
so, x=2π/3 and 4π/3?
anonymous
  • anonymous
yes 4 total. two for the equation \[cos(x)=0\] and two for \[cos(x)=-\frac{1}{2}\]

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