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anonymous

  • one year ago

Find all solutions to the equation. 7 sin2x - 14 sin x + 2 = -5

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  1. mathmate
    • one year ago
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    Not sure if you want to solve: \(7\sin^2(x)-14\sin(x)+2=-5\) or \(7\sin(2x)-14\sin(x)+2=-5\) In either case, factor out the common factor 7 to simplify things a little.

  2. anonymous
    • one year ago
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    sorry the first one

  3. anonymous
    • one year ago
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    But I can't factor out 7 because of the 2

  4. phi
    • one year ago
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    first put it in standard form by adding +5 to both sides \[ 7\sin^2(x)-14\sin(x)+2=-5 \\ 7\sin^2(x)-14\sin(x)+7=0\]

  5. Astrophysics
    • one year ago
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    You can always let sinx = x then factor normally and plug back sinx to find the solution

  6. Astrophysics
    • one year ago
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    \[7x^2-14x+2=5\]

  7. anonymous
    • one year ago
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    I'm sorry. Are you saying all the possible solutions are x=±7 x=±14 and x=±2?

  8. phi
    • one year ago
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    \[ 7\sin^2(x)-14\sin(x)+7=0 \] divide both sides (and all terms) by 7. you get \[ \sin^2(x)-2\sin(x)+1=0 \] to avoid confusion, let \( y= \sin x \) and write this as \[ y^2 -2 y +1= 0 \] can you find the values for y ?

  9. anonymous
    • one year ago
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    I'm so confused. My brother said y=(sqrt-.5)+y

  10. phi
    • one year ago
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    you can factor y^2 -2 y + 1 =0 to get (y-1)(y-1)=0 now solve for y-1= 0

  11. phi
    • one year ago
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    you get y=1 remember y= sin x so sin x =1 what "x" values have sin x = 1?

  12. Astrophysics
    • one year ago
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    Letting x = sinx we get \[7x^2-14x+2=-5 \implies 7x^2-14x+7 =0\] divide both sides by 7 to simplify, we get \[x^2-2x+1=0\] factoring we \[(x-1)^2=0 \implies (x-1)(x-1) = 0\] Now we sub sinx back in to get \[(sinx-1)^2 = 0 \implies (sinx-1)(sinx-1) = 0\] so solving for just sinx -1 = 0, we get sinx = 1

  13. anonymous
    • one year ago
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    So there's only one value?

  14. phi
    • one year ago
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    sin goes forever. between 0 and 2pi (or 0 and 360 degrees) there is only pi/2 (90 degrees) but you can add 2pi to that answer and you will get another value that works people would write the answer as \[ x = \frac{\pi}{2} + k \cdot 2 \pi, \text{ k any integer}\]

  15. anonymous
    • one year ago
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    Thank you so much guys!

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