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Math Help - Half angle or double angle Identities

  1. #1
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    Half angle or double angle Identities

    I don't understand them at all.
    The problem is to evaluate tan(22.5) using exact values.
    How should I do this?
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  2. #2
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    Krizalid's Avatar
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    \tan \frac{x}{2}=\frac{\sin \frac{x}{2}}{\cos \frac{x}{2}}=\frac{\sqrt{\frac{1-\cos x}{2}}}{\sqrt{\frac{1+\cos x}{2}}}=\sqrt{\frac{1-\cos x}{1+\cos x}}.

    Actually is \sin \frac{x}{2}=\pm \sqrt{\frac{1-\cos x}{2}}, but the angle is positive, hence our answer is positive, and that's preserves the positive sign.
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  3. #3
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    Quote Originally Posted by Krizalid View Post
    \tan \frac{x}{2}=\frac{\sin \frac{x}{2}}{\cos \frac{x}{2}}=\frac{\sqrt{\frac{1-\cos x}{2}}}{\sqrt{\frac{1+\cos x}{2}}}=\sqrt{\frac{1-\cos x}{1+\cos x}}.

    Actually is \sin \frac{x}{2}=\pm \sqrt{\frac{1-\cos x}{2}}, but the angle is positive, hence our answer is positive, and that's preserves the positive sign.
    hmm can I assume tan (22.5) is equal to tan 1/2 x?
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  4. #4
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    pickslides's Avatar
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    Krizalid is suggesting x = \frac{\pi}{4}
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  5. #5
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    Well, that was easier than I expected. Thanks.
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