Results 1 to 2 of 2

Math Help - cos^2(36) + sin^2(18)

  1. #1
    Newbie
    Joined
    Sep 2011
    Posts
    5

    cos^2(36) + sin^2(18)

    can you walidate ?

    cos²(36°) + sin²(18°) = 3/4
    Last edited by afrim1; September 24th 2011 at 04:35 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Oct 2010
    From
    Mumbai, India
    Posts
    203

    Re: cos^2(36) + sin^2(18)

    L.H.S. = sin² 18 + cos² 36

    => sin² 18 + ( 1 - 2sin² 18)² - - - - - - (I)

    Now, sin 36 = cos 54 - - {compl. rule }
    => sin 2A = cos 3A - - {Where A = 18 }
    => 2 sin A cos A = 4 cos^3 A - 3 cos A
    => 2 sin A = 4 cos² A - 3 - - - {Divide by cos A }
    => 2 sin A = 4 - 4 sin² A - 3
    => 4 sin² A + 2 sin A - 1 = 0
    => 4 sin² (18) + 2 sin (18) - 1 = 0

    => sin 18 = [-2 ± √(4 + 16) ] / 8 - - - - { By Quadratic Formula }
    => sin 18 = (√5 - 1)/4
    => sin² 18 = [5 + 1 - 2√5] /16
    => sin² 18 = (3 - √5)/8

    Substituting this in - - (i), we get,

    sin² 18 + ( 1 - 2sin² 18)² = (3 - √5)/8 + [1 - 2 (3 - √5)/8]²
    => (3 - √5)/8 + [1 - (3 - √5)/4]²
    => (3 - √5)/8 + [(4 - 3 + √5)/4]²
    => (3 - √5)/8 + [(1 + √5)/4]²
    => (3 - √5)/8 + (1+ 5+ 2√5)/16
    => (3 - √5)/8 + (3 + √5)/8
    => (3 - √5 + 3 + √5)/8
    => 3/4 (Proved)

    reference--- Prove that sin^2 18 + cos^2 36 = 3/4? - Yahoo! Answers India
    Follow Math Help Forum on Facebook and Google+

Search Tags


/mathhelpforum @mathhelpforum