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Math Help - Exact values using trig identities

  1. #1
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    Exact values using trig identities

    How would I find the exact value of sin(pie/16) using trig identities?
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  2. #2
    Behold, the power of SARDINES!
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    You will need the the identities

    \sin\left( \frac{\theta}{2}\right)=\pm \sqrt{\frac{1-\cos(\theta)}{2}}

    \cos\left( \frac{\theta}{2}\right)=\pm \sqrt{\frac{1+\cos(\theta)}{2}}

    Note that

    \sin\left( \frac{\pi}{16}\right)=\sqrt{\frac{1-\cos\left( \frac{\pi}{8}\right)}{2}}

    and

    \cos\left( \frac{\pi}{8}\right)= \sqrt{\frac{1+\cos(\frac{\pi}{4})}{2}}

    I hope this helps
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  3. #3
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    Hello, zweevu!

    We need two Double angle identities:

    . . \sin\theta \:=\:\sqrt{\frac{1-\cos2\theta}{2}}

    . . \cos\theta \:=\:\sqrt{\frac{1+\cos2\theta}{2}}



    Find the exact value of \sin\left(\frac{\pi}{16}\right) using trig identities.

    \sin\left(\frac{\pi}{16}\right) \;=\;\sqrt{\frac{1 - \cos(\frac{\pi}{8})}{2}}

    . . . =\;\sqrt{\frac{1-\sqrt{\dfrac{1 + \cos(\frac{\pi}{4})}{2}}}{2}}

    . . . . =\;\sqrt{\frac{1-\sqrt{\dfrac{1 + \frac{\sqrt{2}}{2} }{2}}}{2}}

    And I'll let you simplify it . . .


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    Well, okay . . . I got: . \frac{\sqrt{2- \sqrt{2 + \sqrt{2}}}}{2}

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  4. #4
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    Well that was simple. Thanks!
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