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Posted by on Monday, December 10, 2007 at 11:36am.

Find the exact value of sin(pi/24)

So like i need like the value like sqrt3/2 but for pi/24...

also if you could show me the steps so i could figure the rest out, that would be cool

  • Math - , Monday, December 10, 2007 at 11:45am

    Use the formula cos(2x) = 2 cos^2(x) - 1

    You can rewrite this as:

    cos(x) = ±sqrt{[cos(2x) + 1]/2}

    Or

    cos(x/2) = ±sqrt{[cos(x) + 1]/2}

    cos(pi/6) = 1/2 sqrt(3)

    Therefore:

    cos(pi/12) = sqrt[1/4 sqrt(3) + 1/2]


    cos(pi/24) =
    sqrt{[sqrt[1/4 sqrt(3) + 1/2] + 1]/2}

    ------------->

    sin(pi/24) = sqrt[1- cos^2(pi/24)]

  • Math - , Monday, December 10, 2007 at 12:00pm

    I need the actual value like 1/2 or sqrt3/2.

  • Math - , Monday, December 10, 2007 at 12:18pm

    The value is in the form of nested square roots as I gave above.

    E.g. there also exists an expression for cos(2 pi/17) and sin(2 pi/17), see here:

    http://en.wikipedia.org/wiki/Heptadecagon#Heptadecagon_construction

    cos(2 pi/n) can be expressed in terms of square roots if n is a power of two times a product of distinct Fermat prime numbers. Fermat prime numbers are prime numbers of the form 2^(2^n) + 1

  • Math - , Monday, December 10, 2007 at 12:27pm

    I was wondering why you were using cosin the problem when the half angle formula for sin is +-sqrt((1-cosx)/2). I can get pi/12, but I can't figure out how to use that answer to get to pi/24

  • Math - , Monday, December 10, 2007 at 12:30pm

    Because the half ange formla of the sin involves the cosine, while the half angle formula for the cosine only involves cosine. The best strategy is thus to compute cos(pi/24) first and then express the sin in terms of the cos using
    sin(pi/24) = sqrt[1-cos^2(pi/24)]

  • Math - , Monday, December 10, 2007 at 12:35pm

    Ok, so say I use the half angle formula in sin. I get:
    sin((pi/6)/2)= sqrt((1-sqrt3/2)/2).

    So sin(pi/12)=sqrt(2-sqrt3)/2
    I get to here by multiplying the whole equation by (2/2).

    So from here, (whew) how to I get to pi/24

  • Math - , Monday, December 10, 2007 at 12:21pm

    I'm confused. I can't seem to make the answer out of what you have.

  • Math - , Monday, December 10, 2007 at 12:28pm

    You just compute cos(x/2) (assumed to be positive) from cos(x) using this formula

    cos(x/2) = sqrt{[cos(x) + 1]/2}

    Take x = pi/6, then cos(pi/6) is known to be 1/2 sqrt(3). The formula gives you cos(pi/12). Then you take x : pi/12, you now now what the cosine is and you work out cos(pi/24). You do all of this symbolically, not numerically, so you get exact expressions.

    Then you express sin(pi/24) as
    sqrt[1-cos^(pi/24)]. You have the symbolic expression for cos(pi/24), so you'll get symbolic expression for
    sin(pi/24)

  • Math - , Monday, December 10, 2007 at 12:28pm

    sin (pi/6)= sin A = = 1/2
    You want the sine of 1/4 of that angle.

    There is a trig identity that says
    sin (1/2)A = sqrt [(1/2)(1 - cos A)]
    and another that says
    cos(1/2)A = sqrt [(1/2)(1 + cos A)]

    sin (pi/12) = sqrt [(1/2)(1- cosA)]
    = sqrt [0.5(1 - (sqrt3)/2)]

    You can apply trigonometric identities one more time to get sin (pi/24), but it gets extremely messy. You will need the value of cos pi/12, which is
    sqrt [0.5(1 + (sqrt3)/2)]

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