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Homework Help: Cos(45 - v) = sin (v + 45) for all angles v?

  1. Oct 11, 2004 #1
    How can I prove that
    cos(45 - v) = sin (v + 45) for all angles v? :uhh:
     
  2. jcsd
  3. Oct 11, 2004 #2
    expand LHS using cos(A-B) formula
    expand RHS using sin(A+B) formula
    show that they are equivalent

    -- AI
     
  4. Oct 11, 2004 #3
    or use the fact that cos(x) = sin(90°-x).

    ofcourse if you wanna prove the above relation you will have to follow to advice of TenaliRaman.

    regards
    marlon
     
  5. Oct 11, 2004 #4
    can one of you show me? I don`t really knowwhere to begin? :shy:
     
  6. Oct 11, 2004 #5

    Zurtex

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    You have, cos(45° - v) = sin (v + 45°)

    Now as said before you should be aware of the relationship, cos(x) = sin(90°-x). All you have to do with this is let x = 45° - v.

    However if you work is in context of the addition of angles then:

    [tex]\sin (A \pm B) = \sin A \cos B \pm \sin B \cos A[/tex]

    [tex]\cos (A \pm B) = \cos A \cos B \mp \sin A \sin B[/tex]

    Let A = 45° and B = v.
     
  7. Oct 11, 2004 #6
    so then I get:
    sin(45+v) = sin 45 cos v + sin v cos 45
    cos(45-v) = cos 45 cos v + sin 45 sin v

    does this prove that cos(45-v) = sin(v+45)?
     
  8. Oct 11, 2004 #7

    Zurtex

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    Almost, what does cos 45° and sin 45° equal?
     
  9. Oct 11, 2004 #8
    0,7071?

    So I don`t have to write more that this?

    I don`t really think I`ve got it yet..
     
    Last edited by a moderator: Oct 11, 2004
  10. Oct 11, 2004 #9

    Zurtex

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    Correct me if I am wrong but both cos 45° and sin 45° are [tex]\frac{\sqrt{2}}{2}[/tex]

    Therefore:

    [tex]\sin (45+v) = \frac{\sqrt{2}}{2} \cos v + \frac{\sqrt{2}}{2} \sin v [/tex]
    [tex]\cos (45-v) = \frac{\sqrt{2}}{2} \cos v + \frac{\sqrt{2}}{2} \sin v[/tex]

    Spot something simmilar? When proving things never ever ever ever ever ever ever ever ever ever ever ever round things off!
     
  11. Oct 12, 2004 #10
    I didn`t know that.. thanks a lot..
     
  12. Oct 12, 2004 #11

    Galileo

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    It's easier with sin(x) = cos(x-90).
    cos(45-v)=cos(v-45) since the cosine is even.
    cos(v-45)=sin(v+90-45)=sin(v+45)
     
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