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Cos2x = 2 cosx sinx

  1. Sep 25, 2004 #1
    Can someone please walk me trough this one:
    cos2x = 2 cosx sinx :uhh:
     
  2. jcsd
  3. Sep 25, 2004 #2

    arildno

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    You are to find those x for which the equation holds (right?)
    What have you thought of thus far?
     
  4. Sep 25, 2004 #3

    mathwonk

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    you might use the fact that 2cos(x)sin(x) = sin(2x) to transform the equation into something easier.
     
  5. Sep 25, 2004 #4

    arildno

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    I would have gotten to that eventually.. :wink:
    Seems like I scared the poster off instead by my questioning..
     
  6. Sep 25, 2004 #5

    HallsofIvy

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    Maria, look closely at what Mathwonk said: your equation has cos(2x)= 2cos(x)sin(x)and you are to find the values of x for which it is true. His equation has sin(2x)= 2cos(x)sin(x) and is an IDENTITY: it is true for all values of x.
     
  7. Sep 25, 2004 #6
    I need to find 4 angles..
    I have found that cos2x = sin2x
    correct?
    But im not sure why?
     
  8. Sep 25, 2004 #7
    Oh sorry.. Mathwonk..I understand.. stupid me..
     
  9. Sep 25, 2004 #8
    I think i`ve got it now
    cos2x = 2cosx sonx
    cos2x = sin2x
    cosx
    sinx = 1
    tanx = 1

    and i get 4 angles because og tan = 1 ?
     
  10. Sep 25, 2004 #9

    arildno

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    You do have cos2x=sin2x
    Dividing with cos2x, you get:
    tan2x=1.
     
  11. Sep 25, 2004 #10
    of course.. forgot..
    then i get
    x =22,5
    this is one angle..
     
  12. Sep 25, 2004 #11

    arildno

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    Certainly; how would you find the others (when restricting 0<x<360)
     
  13. Sep 25, 2004 #12
    180+22,5 = 202,5

    but how about the last two?
     
  14. Sep 25, 2004 #13

    arildno

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    Now, remember that tan(y+180)=tan(y)
    Hence, for any integer n, we have:
    tan(y+180n)=tan(y)
    Set y=45 (i.e, so that tan(y)=1), we may find solutions 0<x360
    by looking at various choices n in the equation:
    2x=45+180n
     
  15. Sep 25, 2004 #14
    So I can for instance set n=1,but thenI get x = 113
    am I right?
     
  16. Sep 25, 2004 #15

    arildno

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    No, you get for n=1: x=22.5+90=112.5
     
  17. Sep 25, 2004 #16
    and for n=-1 I get 292,5
     
  18. Sep 25, 2004 #17
    How du you get from tan(y+180n)=tan(y) to the equation:
    2x=45+180n?
     
  19. Sep 25, 2004 #18

    arildno

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    Well, you're seeking x-solutions satisfying
    tan(2x)=1, or hence:
    tan(2x)=tan(45+180n) for some n
    By setting 2x=45+180n, you're guaranteed the last equation is fulfilled.
     
  20. Sep 26, 2004 #19
    so I don`t need tan since I have it on both sides?
     
  21. Sep 26, 2004 #20

    arildno

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    I hope you have accepted that the x-solutions you're looking for must satisfy:
    tan(2x)=tan(45+180n), where n is some integer (We call this equation (e)).
    Now, to guarantee that (e) holds , requiring 2x=45+180n is evidently enough, since the lefthand side term of (e) (that is tan(2x)) becomes necessarily equal to the righthandside term in (e) (that is tan(45+180n)).
     
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