Mathematics

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1. Show that Sin3è - Cos2è = (1 - Sinè)(4Sin^2è + 2Sinè - 1).
Without using a calculator, show that è = 18 degrees is an exact solution of the equation Sin3è = Cos2è.
Justifying your answer, find the exact values of:
i) Sin18 degrees.
ii) Sin234 degrees.

  • Mathematics -

    I will use x instead of è

    sin 3x - cos 2x = (1-sinx)(4sin^2 x + 2sinx - 1)
    RS = 4sin^2 x + 2sinx - 1 - 4sin^3 x + 2sin^2 x + sinx
    = -4sin^3 x + 2sin^2 x + 3sinx - 1

    LS = sin(2x+x) - cos 2x
    = (sin 2x)(cosx) + (cos 2x)(sinx) - (1 - 2sin^2 x)
    = 2sinxcosxcosx + sinx(1-2sin^2 x) - 1 + 2sin^2 x
    = 2sinxcos^2 x + sinx - 2sin^3 x - 1 + 2sin^2 x
    = 2sinx(1 - sin^2 x) + sinx - 2sin^3 x - 1 + 2sin^2 x
    = 2sinx - 2sin^3 x + sinx - 2sin^3 x - 1 + 2sin^2 x
    = -4sin^3 x + 2sin^2 x + 3sinx - 1
    = RS

    For your next question, show that 18° is a solution to
    sin 3x = cos 2 or
    sin 3x - cos 2x = 0
    since I expanded this expression in LS above
    we have to solve
    -4sin^3 x + 2sin^2 x + 3sinx - 1 = 0 or
    4sin^3 x - 2sin^2 x - 3sinx + 1 = 0
    let sinx = y
    4y^3 - 2y^2 - 3y + 1 = 0
    clearly y = 1 is a solution
    (y-1)(4y^2 + 2y -1) = 0

    the other roots are (-1 ± √5)/4
    so sinx = (-1 ± √5)/4

    This also answers your second-last question.
    that sin 18° = (√5 - 1)/4


    I now have to show that this equals sin 18° without a calculator, (using my calculator shows me that so far I am correct, since 18 would be a solution using my machine)

    Working on this ......

  • Mathematics -

    sin 18° = cos 72° by the complementary property.

    going out on a far-fetched limb here .......
    draw a pentagon ABCDE
    draw diagonals AC and BD to intersect at P
    Look at triangle ABC, angle B = 108 and angles A and C are 36° each.
    If AB = 1, I happen to know that the diagonal : side = the golden ratio
    which is (1 + √5)/2 : 1

    so let AB = 2, then AC = √5 + 1
    using the cosine law in triangle ABC
    2^2 = 2^2 + (√5+1)^2 – 2(2)(√5+1)cos 36°
    4 = 4 + 5 + 2√5 + 1 – 4(√5+1)cos 36
    cos 36 = (6+2√5)/(4(√5+1))
    = (√5 + 1)/4 after rationalizing the denominator.

    Now cos 72 = 2cos^2 36° - 1 , using cos 2A = 2cos^2 A - 1
    = 2[(√5+1)/4]^2 – 1
    = 2(5 + 2√5 + 1)/16 – 1
    = (12 + 4√5)/16 – 16/16
    = (-4 + 4√5)/16
    = (-1 + √5)/4

    but sin 18 = cos 72, as noted above

    so sin 18° = (-1 + √5)/4

  • Mathematics -

    for your last one:

    234 = 180 + 54
    and 54 = half of 108 which is the large angle in the pentagon.

    sin234 = sin(180 + 54) = sin180cos54 + cos180sin54
    = 0 - sin54
    = -sin 54°

    you can find cos 108° using the cosine law in triangle ABC
    then
    cos 108 = 1 - 2sin^2 54°

    give it a shot.

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