Math

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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 -

    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 -

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

  • Math -

    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 -

    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 -

    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 -

    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 -

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

  • Math -

    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 -

    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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