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Math Help - trigonometry

  1. #1
    Junior Member Singular's Avatar
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    trigonometry

    How to proof ,

    <br />
(\sin x + \cos x)\sin 2x = \sqrt 2 <br />

    thanks
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  2. #2
    Math Engineering Student
    Krizalid's Avatar
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    You probably meant to say "how to solve" instead "how to prove," since as it's written, it just holds for certain values, in fact, it's a trig. equation.
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  3. #3
    Member Last_Singularity's Avatar
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    Quote Originally Posted by Singular View Post
    How to proof ,

    <br />
(\sin x + \cos x)\sin 2x = \sqrt 2 <br />

    thanks
    (\sin x + \cos x)\sin 2x = \sqrt 2 cannot be proved because it is not true. I think that what you meant to ask is how to solve that equation for x

    Try squaring both sides and expanding:
    (\sin x + \cos x)^2 \sin^2 2x = 2
    (\sin^2 x + 2 \cos x \sin x + \cos^2 x) \sin^2 2x = 2
    (1+ 2 \cos x \sin x) \sin^2 2x = 2
    (1+ \sin 2x) \sin^2 2x = 2

    Letting y = \sin 2x, you have
    (1+ \sin 2x) \sin^2 2x = 2 \longrightarrow (1+y)y^2 = 2

    Solve for y by inspection. I see one obvious value of y such that (1+y)y^2 = 2

    Then find when \sin 2x equals that value of y and you're done.

    Note: there is more than one solution. In fact, there are infinitely many solutions to this problem because sine is a periodic function.
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  4. #4
    Junior Member Singular's Avatar
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    owh... my mistake... sorry
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