Results 1 to 5 of 5

Math Help - I need help

  1. #1
    Junior Member
    Joined
    Jul 2008
    Posts
    47

    I need help

    Use identities to find each exact value. (Do not use a calculator).


    cos(-pi/12)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Hello,

    Quote Originally Posted by norwoodjay View Post
    Use identities to find each exact value. (Do not use a calculator).


    cos(-pi/12)
    We know that cos(-x)=cos(x)

    Thus cos(-pi/12)=cos(pi/12).
    pi/12=(pi/6)/2.
    --> cos(-pi/12)=cos((1/2) pi/6)

    We know that cos(2x)=2cosē(x)-1. Then cos(x)=2cosē(x/2)-1 --> cos(x/2)=+ or - sqrt[(cos(x)+1)/2]

    therefore cos(-pi/12)=+ or - sqrt[(cos(pi/6)+1)/2]
    Since -pi/12 is in the fourth quadrant, the cosine is positive.

    --------> cos(-pi/12)=sqrt[(cos(pi/6)+1)/2]

    (and you should know cos(pi/6))

    cos(-pi/12)=sqrt(sqrt(3)+2)/2
    Last edited by Moo; July 26th 2008 at 08:27 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Jun 2008
    Posts
    792
    cos(-pi/12) = cos(pi/12)

    Why? Because cosine is an even function. Now, let's rewrite pi/12 as:
    pi/4 - pi/6

    So we got now cos(pi/12) = cos(pi/4 - pi/6). Now, recall the sum and difference identites for cosine:

    cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
    cos(A - B) = cos(A)cos(B) + sin(A)sin(B)

    Let's use it then!

    cos(pi/4 - pi/6) = cos(pi/4)cos(pi/6) + sin(pi/4)sin(pi/6)

    Now, since they asked you not to use the calculator, you must use the special triangles
    1. pi/4 - pi/4 - pi/2 (45-45-90)
    2. pi/3 - pi/2 - pi/6 (60-90-30)

    cos(pi/4 - pi/6) = ( (sqrt(2)/2) x (sqrt(3)/2) ) + ( (sqrt(2)/2) x 1/2 )
    = (sqrt(6)/4) + (sqrt(2)/4)

    Final answer: ( sqrt(6) + sqrt(2) )/4
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Note that our solutions are equal !

    ( sqrt(6) + sqrt(2) )/4

    sqrt(sqrt(3)+2)/2


    See in this post of earboth the identity : http://www.mathhelpforum.com/math-help/32751-post3.html (the identity can easily be shown by squaring both sides)
    Follow Math Help Forum on Facebook and Google+

  5. #5
    A riddle wrapped in an enigma
    masters's Avatar
    Joined
    Jan 2008
    From
    Big Stone Gap, Virginia
    Posts
    2,551
    Thanks
    12
    Awards
    1
    Quote Originally Posted by norwoodjay View Post
    Use identities to find each exact value. (Do not use a calculator).


    cos(-pi/12)
    Follow Math Help Forum on Facebook and Google+


/mathhelpforum @mathhelpforum