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Math Help - Double Identity Problems

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    Double Identity Problems

    I know it has something to do with the double identities, but I don't know where to start. Thanks in advance.

    If x is a positive acute angle and  sin  x = \frac{1}{2} , what is  sin 2x ?
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    Quote Originally Posted by ThaBeretta View Post
    I know it has something to do with the double identities, but I don't know where to start. Thanks in advance.

    If x is a positive acute angle and  sin x = \frac{1}{2} , what is  sin 2x ?
    You need to know that sin(2x) = 2 sinx cos x (this is one of the famous double angle formulas).

    You're given sin x so now all you have to do is calculate cos x. I assume this will be similar to problems you have previously done.

    Then substitute both into the above formula.
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    Quote Originally Posted by mr fantastic View Post
    You need to know that sin(2x) = 2 sinx cos x (this is one of the famous double angle formulas).

    You're given sin x so now all you have to do is calculate cos x. I assume this will be similar to problems you have previously done.

    Then substitute both into the above formula.
    Hint for finding \cos{x}:

    Remember another famous identity, \sin^2{x} + \cos^2{x} = 1.
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    Quote Originally Posted by mr fantastic View Post
    You need to know that sin(2x) = 2 sinx cos x (this is one of the famous double angle formulas).

    You're given sin x so now all you have to do is calculate cos x. I assume this will be similar to problems you have previously done.

    Then substitute both into the above formula.
    I assumed I had to use that identity because it's the only one that starts with sin 2x, but how would I calculate the value of cos x if there are two unknowns:

    sin2x = 2sinx cosx

    sin2x = 2 (1/2) cos x

    sin2x = cosx
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    Quote Originally Posted by ThaBeretta View Post
    I assumed I had to use that identity because it's the only one that starts with sin 2x, but how would I calculate the value of cos x if there are two unknowns:

    sin2x = 2sinx cosx

    sin2x = 2 (1/2) cos x

    sin2x = cosx
    You get the value of cosx from the known fact that sin x = 1/2 and x is in the first quadrant.

    You then substitute the values of sin x and cos x into 2 sin x cos x to get the value of sin(2x).
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    Given that information, cos x must then equal radical 3/2, correct? So the final answer is radical 3/2 (sin x is 1/2 and that is multiplied by 2, giving you 1).
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    Quote Originally Posted by ThaBeretta View Post
    Given that information, cos x must then equal radical 3/2, correct? So the final answer is radical 3/2 (sin x is 1/2 and that is multiplied by 2, giving you 1).
    Yes.

    Another way of doing it would be to note that if sin x = 1/2 and x lies in the first quadrant, then x = pi/6. Hence 2x = pi/3 and so sin(2x) = sin(pi/3).
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