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Thread: Formulas Involving Nested Radicals - Trig: (sqrt(2) - sqrt(6))/4 = (-2sqrt(2-sqrt(3))

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    Formulas Involving Nested Radicals - Trig: (sqrt(2) - sqrt(6))/4 = (-2sqrt(2-sqrt(3))

    Hello all,

    Currently, we are covering the trigonometric identities (half-angle, sum, difference, etc.). One particular problem asked to solve for the sin(195deg). So, I first attempted with the sum formula, splitting the 195 into 150 and 45. This yielded the result (sqrt(2) - sqrt(6))/4. However, I then wondered what the result would be utilizing the half-angle formula. Using this, I found the answer to be (-2sqrt(2-sqrt(3))/4. I then set these answers equal to each other and found that -2sqrt(2-sqrt(3)) = sqrt(2) - sqrt(6). Therefore, is there a formula involving nested radicals (like the one found in the half-angle result) that says, for example, what sqrt(A - sqrt(B)) is equivalent to? I am unsure of the correlation between the two numbers on either side of the equation, so any starter help is greatly appreciated.

    Thank you!

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    In this particular case we can prove the equality:
    sqrt(2) - sqrt(6) = -1*(sqrt(6) - sqrt(2)) = - sqrt( (sqrt(6) - sqrt(2))2 ) = - sqrt( 6 - 2*sqrt(6)*sqrt(2) + 2 )​=
    = - sqrt( 8 - 2*sqrt(12) )​ = - sqrt( 8 - 4*sqrt(3) )​= - sqrt( 4(2 - sqrt(3)) ) = -2 sqrt( 2 - sqrt(3) )

    I am not aware of a general case.

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    Quote Originally Posted by austint View Post
    Hello all,

    Currently, we are covering the trigonometric identities (half-angle, sum, difference, etc.). One particular problem asked to solve for the sin(195deg). So, I first attempted with the sum formula, splitting the 195 into 150 and 45. This yielded the result (sqrt(2) - sqrt(6))/4. However, I then wondered what the result would be utilizing the half-angle formula. Using this, I found the answer to be (-2sqrt(2-sqrt(3))/4. I then set these answers equal to each other and found that -2sqrt(2-sqrt(3)) = sqrt(2) - sqrt(6). Therefore, is there a formula involving nested radicals (like the one found in the half-angle result) that says, for example, what sqrt(A - sqrt(B)) is equivalent to? I am unsure of the correlation between the two numbers on either side of the equation, so any starter help is greatly appreciated.

    Thank you!
    UPDATE: So, I read up on "de-nesting" of radicals and discovered a pattern which read sqrt(X-Y) = sqrt(X) - sqrt(Y). Therefore, the solution to this question is what would sqrt(X - sqrt(Y)) equal to.

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    Quote Originally Posted by austint View Post
    UPDATE: So, I read up on "de-nesting" of radicals and discovered a pattern which read sqrt(X-Y) = sqrt(X) - sqrt(Y). Therefore, the solution to this question is what would sqrt(X - sqrt(Y)) equal to.
    Huh? This is not true, in general.

    E.g. sqrt(49-25) = sqrt(24) = ~4.9

    sqrt(49) - sqrt(25) = 7-5 = 2 != 4.9

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    Quote Originally Posted by j-astron View Post
    Huh? This is not true, in general.

    E.g. sqrt(49-25) = sqrt(24) = ~4.9

    sqrt(49) - sqrt(25) = 7-5 = 2 != 4.9
    Yes, you're entirely right here. I mistyped the formula here. This is what I found: If sqrt[a + sqrt(b)] = sqrt(x) + sqrt(y),
    where a, b, x, and y are rational expressions, and
    a greater than sqrt(b), then
    sqrt[a - sqrt(b)] = sqrt(x) - sqrt(y).

    However, I'm not sure how to derive this.

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    Quote Originally Posted by austint View Post
    Yes, you're entirely right here. I mistyped the formula here. This is what I found: If sqrt[a + sqrt(b)] = sqrt(x) + sqrt(y),
    where a, b, x, and y are rational expressions, and
    a greater than sqrt(b), then
    sqrt[a - sqrt(b)] = sqrt(x) - sqrt(y).

    However, I'm not sure how to derive this.
    I haven't proved it either, but I think I found a counterexample

    a = 16
    b = 81
    x = 4
    y = 9

    [tex] \sqrt{a + \sqrt{b}} = \sqrt{16 + \sqrt{81}} = \sqrt{16 + 9} = \sqrt{25} = 5[/tex]

    [tex] \sqrt{x} + \sqrt{y} = \sqrt{4} + \sqrt{9} = 2+3 = 5 [/tex]

    So the first condition is satisfied. Does the statement hold?

    [tex] \sqrt{a - \sqrt{b}} = \sqrt{16 - \sqrt{81}} = \sqrt{16 - 9} = \sqrt{7} \approx 2.646[/tex]

    [tex] \sqrt{x} - \sqrt{y} = \sqrt{4} - \sqrt{9} = 2-3 = -1 [/tex]

    These are not equal to each other. Were there additional conditions that had to be true for the statement to hold?

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    Quote Originally Posted by j-astron View Post
    I haven't proved it either, but I think I found a counterexample

    a = 16
    b = 81
    x = 4
    y = 9

    [tex] \sqrt{a + \sqrt{b}} = \sqrt{16 + \sqrt{81}} = \sqrt{16 + 9} = \sqrt{25} = 5[/tex]

    [tex] \sqrt{x} + \sqrt{y} = \sqrt{4} + \sqrt{9} = 2+3 = 5 [/tex]

    So the first condition is satisfied. Does the statement hold?

    [tex] \sqrt{a - \sqrt{b}} = \sqrt{16 - \sqrt{81}} = \sqrt{16 - 9} = \sqrt{7} \approx 2.646[/tex]

    [tex] \sqrt{x} - \sqrt{y} = \sqrt{4} - \sqrt{9} = 2-3 = -1 [/tex]

    These are not equal to each other. Were there additional conditions that had to be true for the statement to hold?
    I found the statement in Webster Well's "Advanced Course in Algebra" and included that a, b, x, y are rational expressions and a is greater than sqrt(b). However, I believe that verifying the statement only proves that it is true for this qualifications. I want to find actual values of x and y in relation to a and b. Possibly, I can find more equalities by solving more half-angle and double-angle trig identities and then finding a relation between the numbers?

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    Cool

    Quote Originally Posted by austint View Post
    One particular problem asked to solve for the sin(195deg). So, I first attempted with the sum formula, splitting the 195 into 150 and 45. This yielded the result:

    . . . . .[tex]\dfrac{\sqrt{2\,}\, -\, \sqrt{6\,}}{4}[/tex]

    However, I then wondered what the result would be utilizing the half-angle formula. Using this, I found the answer to be:

    . . . . .[tex]\dfrac{-2\, \sqrt{2\, -\, \sqrt{3\,}\,}}{4}[/tex]

    I then set these answers equal to each other and found that -2sqrt(2-sqrt(3)) = sqrt(2) - sqrt(6). Therefore, is there a formula involving nested radicals (like the one found in the half-angle result) that says, for example, what sqrt(A - sqrt(B)) is equivalent to?
    There is no formula, that I'm aware of, but there is a methodology.

    First let's note that we only really care about the numerators, since the denominators are the same. Also, it would be nicer to deal with non-negative values, so let's work with these:

    . . . . .[tex]\sqrt{6\,}\, -\, \sqrt{2\,}\, \mbox{ and }\, 2\, \sqrt{2\, -\, \sqrt{3\,}\,}[/tex]

    In particular, we would like to show that the right-hand expression is equal to the left-hand expression. We'll assume that the right-hand expression can be stated in terms of a difference of radicals:

    . . . . .[tex]2\, \sqrt{2\, -\, \sqrt{3\,}\,}\, =\, \sqrt{a\,}\, -\, \sqrt{b\,}[/tex]

    Square both sides:

    . . . . .[tex]4\, (2\, -\, \sqrt{3\,})\, =\, a\, -\, 2\, \sqrt{ab\,}\, +\, b[/tex]

    . . . . .[tex]8\, -\, 4\, \sqrt{3\,}\, =\, (a\, +\, b)\, -\, 2\, \sqrt{ab\,}[/tex]

    Then:

    . . . . .[tex]8\, =\, a\, +\, b[/tex]

    . . . . .[tex]4\, \sqrt{3\,}\, =\, 2\, \sqrt{ab\,}[/tex]

    The second equation above gives us:

    . . . . .[tex]\sqrt{12\,}\, =\, \sqrt{ab\,}[/tex]

    . . . . .[tex]12\, =\, ab[/tex]

    . . . . .[tex]\dfrac{12}{a}\, =\, b[/tex]

    Plug this into:

    . . . . .[tex]8\, =\, a\, +\, b[/tex]

    ...and see where this leads....

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