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Pe: 2015-09-05T07:50:10+03:00 In: In:

Trigonometrie

1.Sa se transforme in produs:
a)sinx+siny+sin(x+y)
b)sin(x+z)+sin(y-z)-sin(x+y)
c)sinx+siny-sin(x+y)
d)sin(x+z)+sin(y-z)-sin(x+y)
2.Sa se transforme in produs
a)1+cosx+cosy+cos(x+y)
b)1+cos(x+z)+cos(y-z)+cos(x+y)
c)cosx-cosy+sin(x+y).
La punctul 1 am incercat toate variantele am combinat prima cu ultima si am ajuns tot la o adunare la care nu puteai scoate nimic factor comum.
La punctul 2 la fel am facut si nu am ajuns unde trebuia la 4cosx+y/2.cosx/2+cosy

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7 raspunsuri

  1. adrianS

    1)Pentru punctul 1a scriem:
    sinx+siny+2sin\dfrac{x+y}{2}cos\dfrac{x+y}{2}=2sin\dfrac{x+y}{2}cos\dfrac{x-y}{2}+2sin\dfrac{x+y}{2}cos\dfrac{x+y}{2}=\\ 2sin\dfrac{x+y}{2}(cos\dfrac{x-y}{2}+cos\dfrac{x+y}{2})=2sin\dfrac{x+y}{2}(2cos\dfrac{\dfrac{x-y}{2}+\dfrac{x+y}{2}}{2}cos\dfrac{\dfrac{x-y}{2}-\dfrac{x+y}{2}}{2})=\\ 4sin\dfrac{x+y}{2}cos\dfrac{x}{2}cos\dfrac{y}{2}
    Celelalte se fac asemanator.Daca nu va descurcati va rog sa imi spuneti sa le facem impeuna.

  2. pp

    Dar de ce ai scris sin(x+y) ca 2 sinx+y/2.cosx+y/2?

  4. adrianS

    Este relatia sinusului unghiului dublu:
    sin2\alpha=2sin\alpha\cdot cos\alpha\\ sin\alpha=2sin\dfrac{\alpha}{2}cos\dfrac{\alpha}{2}

  5. pp

    Multumesc!

  6. pp

    Ma ajuti putin si la 2.Am incercat sa scriu cos(x+y)=2cos^2a/2-sin^2a/2 dar nu ajung la nici un factor comun.

  8. adrianS

    2)Pai scriem:
    1+cosx+cosy+cos(x+y)=1+cosx+cosy+cosxcosy-sinxsiny=\\ =(1+cosx)(1+cosy)-sinxsiny=(cos0^0+cosx)(cos0^0+cosy)-sinxsiny=\\ =2cos^2\dfrac{x}{2}\cdot 2cos^2\dfrac{y}{2}-sinxsiny=4cos^2\dfrac{x}{2}cos^2\dfrac{y}{2}-sinxsiny=4cos^2\dfrac{x}{2}cos^2\dfrac{y}{2}-4sin\dfrac{x}{2}cos\dfrac{x}{2}sin\dfrac{y}{2}cos\dfrac{y}{2}=\\ =4cos\dfrac{x}{2}cos\dfrac{y}{2}(cos\dfrac{x}{2}cos\dfrac{y}{2}-sin\dfrac{x}{2}sin\dfrac{y}{2})=\\ =4cos\dfrac{x}{2}cos\dfrac{y}{2}cos(\dfrac{x}{2}-\dfrac{y}{2}
    Daca mai aveti probleme si pentru altele,rog sa ma contactati.

  9. ghioknt
    profesor

    Pentru transformarea rapidă a unor sume în produse sunt utile 2 mici taine ale trigonometriei:
    1+\cos t=2\cos^2\frac{t}{2};\;\;\;1-\cos t=2\sin^2\frac{t}{2}.
    Astfel, 1+\cos(x+y)+\cos x+\cos y=2\cos^2\frac{x+y}{2}+2\cos\frac{x+y}{2}\cos\frac{x-y}{2}=\\=2\cos\frac{x+y}{2}\left ( \cos\frac{x+y}{2}+\cos\frac{x-y}{2} \right )=4\cos\frac{x+y}{2} \cos\frac{x}{2}\cos\frac{y}{2}.

    1+\cos(x+y)+\cos(x+z)+\cos(y-z)=2\cos^2\frac{x+y}{2}+2\cos\frac{x+y}{2}\cos\left ( \frac{x-y}{2}+z \right )=\\=2\cos\frac{x+y}{2}\left ( \cos\frac{x+y}{2}+\cos\left ( \frac{x-y}{2}+z \right ) \right )=4\cos\frac{x+y}{2}\cos\frac{x+z}{2}\cos\frac{y-z}{2}.

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