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Math Help - simplify and proof that answer is this

  1. #1
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    simplify and proof that answer is this

    can anybody help?

    simplify and proof that answer is this-matttte.jpg
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    Forum Admin topsquark's Avatar
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    Re: simplify and proof that answer is this

    Quote Originally Posted by tautvyduks View Post
    can anybody help?

    Click image for larger version. 

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    For the first problem, note that
    sin(A - B) = sin(A)~cos(B) - sin(B)~cos(A)

    so we have:
    sin \left ( \frac{\pi}{3} - \alpha \right ) = sin \left ( \frac{\pi}{3} \right ) ~cos( \alpha ) - sin( \alpha) ~cos \left ( \frac{\pi}{3} \right )

    See if that helps.

    -Dan
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    MHF Contributor MarkFL's Avatar
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    Re: simplify and proof that answer is this

    How do you begin with an expression and end with an equation?
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    Re: simplify and proof that answer is this

    but what i should do with 1/2sin(alpha)
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    Re: simplify and proof that answer is this

    Subtract it from the expression topsquark gave you after you have simplified it.
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    Forum Admin topsquark's Avatar
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    Re: simplify and proof that answer is this

    For the second problem, note that sin(2x) = 2~sin(x)~cos(x), so we have
    1 - sin^4(x) - sin^2(x) = \frac{1}{4}sin^2(x)

    1 - sin^4(x) - sin^2(x) = \frac{1}{4} \left ( 2~sin(x)~cos(x) \right ) ^2

    Multiply out the RHS and use cos^2(x) = 1 - sin^2(x), then solve for sin(x).

    You will get sin^2(x) = \frac{1}{2}

    Note that there will be four solutions to this.

    -Dan

    Wow. Lots of posting going on in this thread!
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  7. #7
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    Re: simplify and proof that answer is this

    x
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  8. #8
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    Re: simplify and proof that answer is this

    ok, thanks i did that but what i have to do with -1/2sin(alpha)? im bad at maths sorry
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  9. #9
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    Re: simplify and proof that answer is this

    Quote Originally Posted by topsquark View Post
    For the second problem, note that sin(2x) = 2~sin(x)~cos(x), so we have
    1 - sin^4(x) - sin^2(x) = \frac{1}{4}sin^2(x)

    1 - sin^4(x) - sin^2(x) = \frac{1}{4} \left ( 2~sin(x)~cos(x) \right ) ^2

    Multiply out the RHS and use cos^2(x) = 1 - sin^2(x), then solve for sin(x).

    You will get sin^2(x) = \frac{1}{2}

    Note that there will be four solutions to this.

    -Dan

    Wow. Lots of posting going on in this thread!
    Thanks, but i need to solve this i need to proof that
    1 - sin^4(x) - sin^2(x)              is equal to              frac{1}{4}sin^2(x)

    i
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  10. #10
    MHF Contributor MarkFL's Avatar
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    Re: simplify and proof that answer is this

    The second problem is:

    1 - \sin^4(x) - \sin^2(x) = \frac{1}{4}\sin^2(2x)

    This is not an identity to prove, but an equation to solve.

    Using the double-angle identity for sine provided by topsquark, we have:

    1 - \sin^4(x) - \sin^2(x) = \frac{1}{4}(2\sin(x)\cos(x))^2

    1 -\sin^4(x) - \sin^2(x) = \sin^2(x)\cos^2(x)

    1 - \sin^4(x) - \sin^2(x) = \sin^2(x)(1-\sin^2(x))

    1 - \sin^4(x) - \sin^2(x) = \sin^2(x)-\sin^4(x)

    1 - \sin^2(x) = \sin^2(x)

    1 = 2\sin^2(x)

    \sin^2(x)=\frac{1}{2}

    \sin(x)=\pm\frac{1}{\sqrt{2}}

    You will find 4 solutions on the interval 0\le x<2\pi. If you are not restricted to an interval, then you will have an infinite number of solutions, which is easily expressed as the integral multiple of an angle.
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  11. #11
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    Re: simplify and proof that answer is this

    i need to suplify left side of this and get the right side.. not to solve X...
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  12. #12
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    Re: simplify and proof that answer is this

    It is not an identity. It is only true for certain values of x.
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  13. #13
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    Re: simplify and proof that answer is this

    ohhh it`s imposibble tu supplify it am i right?
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  14. #14
    MHF Contributor MarkFL's Avatar
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    Re: simplify and proof that answer is this

    It is impossible to take the left side and get the right, yes.
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  15. #15
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    Re: simplify and proof that answer is this

    oh thanks. and can you help with the 1st one?
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