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anonymous

  • one year ago

Will METAL!! Verify the identity. cos (x - y) - cos (x + y) = 2 sin x sin y

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  1. Vocaloid
    • one year ago
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    we'll be using two rules here: cos(A+B) = cosA*cosB - sinA*sinB cos(A-B) = cos(A)*cos(B) + sin(A)*sin(B)

  2. Vocaloid
    • one year ago
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    cos(x-y) = cosx*cosy + sinx*siny cos(x+y) = cosx*cosy - sinx*siny therefore: cos(x-y) - cos(x+y) = cosx*cosy + sinx*siny - (cosx*cosy - sinx*siny) = 2sinx*siny let me know if anything needs to be clarified

  3. anonymous
    • one year ago
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    wait could you break that down a bit.

  4. Vocaloid
    • one year ago
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    sure. start with cos (x - y) = cosx*cosy + sinx*siny is that clear?

  5. anonymous
    • one year ago
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    yes

  6. Vocaloid
    • one year ago
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    cos (x + y) = cosx*cosy - sinx*siny is that clear?

  7. anonymous
    • one year ago
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    yes

  8. Vocaloid
    • one year ago
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    good, now we just subtract cos(x-y) - cos(x+y) = cosx*cosy + sinx*siny - (cosx*cosy - sinx*siny) is this part clear?

  9. anonymous
    • one year ago
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    alright what's next after that?

  10. Vocaloid
    • one year ago
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    just basic algebra after that

  11. Vocaloid
    • one year ago
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    notice how the cosines cancel out

  12. Vocaloid
    • one year ago
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    |dw:1443824395082:dw|

  13. Vocaloid
    • one year ago
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    |dw:1443824403678:dw|

  14. anonymous
    • one year ago
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    oh ok, could we do. Using the given zero, find all other zeros of f(x). -2i is a zero of f(x) = x4 - 21x2 - 100

  15. Vocaloid
    • one year ago
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    @freckles having a bit of a brain fart about complex zeros

  16. anonymous
    • one year ago
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    lol ok thats fine. one min brb

  17. Vocaloid
    • one year ago
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    I remember that complex zeros come in pairs, so if -2i is a zero, then +2i must also be a zero

  18. Vocaloid
    • one year ago
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    so, we can write an expression using each root x + 2i = 0 x - 2i = 0 then I'm guessing we multiply these two together to get (x+2i)(x-2i) = 0 x^2 - (2i)^2 = x^2 - (-4) = x^2 + 4 as our polynomial

  19. Vocaloid
    • one year ago
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    then we take our original function f(x) = x4 - 21x2 - 100 divide by the polynomial we just got x^2 + 4 using synthetic division, I presume

  20. Vocaloid
    • one year ago
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    |dw:1443824947058:dw| (it's been so long since I've done this lol)

  21. Vocaloid
    • one year ago
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    |dw:1443825019233:dw|

  22. Vocaloid
    • one year ago
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    |dw:1443825082485:dw|

  23. Vocaloid
    • one year ago
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    that leaves us with x^2 - 25 and the roots of that polynomial are 5 and -5

  24. Vocaloid
    • one year ago
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    anyway, that means our missing roots are 5 and -5, would you like me to go over anything with you?

  25. Vocaloid
    • one year ago
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    hopefully I didnt make a mistake somewhere

  26. anonymous
    • one year ago
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    @Vocaloid sorry I lost connection.

  27. anonymous
    • one year ago
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    For the given function, find the vertical and horizontal asymptote(s) (if there are any). f(x) = the quantity x squared plus three divided by the quantity x squared minus nine x = 3, x = -3, y = 1 x = 9, y = 3 None x = 9, y = 1

  28. anonymous
    • one year ago
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    options^

  29. anonymous
    • one year ago
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    |dw:1443825614299:dw|

  30. Vocaloid
    • one year ago
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    hint: to find the vertical asymptote, set the denominator equal to 0

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