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Math Help - Trignometric Functions

  1. #1
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    Exclamation Trignometric Functions

    Can somebody help me with these?

    1. Simplify the expression sinx (cscx - sin x)
    2. rewrite the expression cos^2 / (1 + sin x) so that there is no fraction
    3. use the half-angle formula to simplify the expression sqrt ((1+cos(4x))/2)
    4. Use the following to find the exact value of sin(x/2). cosx = -4/5, x is in quadrant III
    5. Evaluate the expression sin^2(355pi/ 113) + cos^2 (355pi/113)
    6. verify the identity cos x/(1 + sinx) + cosx/(1-sinx) = 2 secx

    Thank you!
    Isabel
    Last edited by cuteisa89; May 13th 2007 at 04:11 PM.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by cuteisa89 View Post
    1. Simplify the expression sinx (cscx - sin x)
    sinx(cscx - sinx) = sinx(1/sinx - sinx)
    .......................= sinx/sinx - sinx*sinx
    .......................= 1 - sin^2x
    .......................= cos^2x

    2. rewrite the expression cos^2 / (1 + sin x) so that there is no fraction
    cos^2x/(1 + sinx) = (1 - sin^2x)/(1 + sinx) ...the top is the difference of two squares
    .........................= [(1 + sinx)(1 - sinx)]/(1 + sinx)
    .........................= 1 - sinx


    3. use the half-angle formula to simplify the expression sqrt ((1+cos(4x))/2)
    by the half angle formula:

    cos(x/2) = sqrt[(1 + cosx)/2]

    if we replace x with 4x we get:

    cos(2x) = sqrt[(1 + cos(4x))/2]

    so cos(2x) is the simplified expression

    4. Use the following to find the exact value of sin(x/2). cosx = -4/5, x is in quadrant III
    sin(x/2) = - sqrt[(1 - cosx)/2] ..........sine is negative in quad III
    we are given that cosx = -4/5
    => sin(x/2) = -sqrt[(1 - (-4/5))/2]
    ................= -sqrt[(9/5)/2]
    ................= -sqrt(9/10)
    ................= -3/sqrt(10)
    ................= -3sqrt(10)/10


    5. Evaluate the expression sin^2(355pi/ 113) + cos^2 (355pi/113)
    sin^2(355pi/ 113) + cos^2 (355pi/113) = 1
    since sin^2x + cos^2x = 1 for all x


    6. verify the identity cos x/(1 + sinx) + cosx/(1-sinx) = 2 secx
    Tips for tackling such problems:
    1) start on the most complicated side, it gives you more options to change things
    2) it is USUALLY better to change everything to sines and cosines. these are the ones students are more familiar with, so it's easier to see connections
    3) always consider working on both sides one at a time to bring them to the same thing. sometimes working on one side only to get the answer is too hard or even impossible in some sense

    see below for solution
    Attached Thumbnails Attached Thumbnails Trignometric Functions-identity.gif  
    Last edited by Jhevon; May 13th 2007 at 07:11 PM.
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