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Solve for X as shown in the below sketch

  1. Jan 3, 2014 #1
    1. The problem statement, all variables and given/known data
    As shown in the attached sketch, derive a non-transcendental equation for the angle, X, given that the parameters A,B, d and R are known variables.

    2. Relevant equations

    Basic trig.

    3. The attempt at a solution
    I have derived an equation which is a function of X as shown below based on simple trignometry:

    B = d + R + Rcos(x) + dcos(x) + (A-d-R)sin(x)

    Which can be arranged into a (somewhat) easier format:

    (R+d)cos(x) + (A-d-R)sin(x) = B - d - R

    However I cannot simplify this expression any further. If anyone can help me with this I would really appreciate it! I hope I have been clear in showing the problem geometry, any ambiguities please let me know. Thank you so much.
     

    Attached Files:

  2. jcsd
  3. Jan 3, 2014 #2

    mfb

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    Every sum of A*cos(x)+B*sin(x) can be combined to a single function like C*sin(x+D). Those identities help to calculate C and D.
     
  4. Jan 3, 2014 #3

    SammyS

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    Look at the "Linear Combination" section of "List of trigonometric identities" in Wikipedia by using the following link.

    http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Linear_combinations

    Essentially, ##\displaystyle A\sin x+B\cos x=\sqrt{A^2+B^{\,2}}\cdot\sin(x+\varphi) ##

    where ##\displaystyle\ \varphi = \arctan \left(\frac{B}{A}\right) + \begin{cases}
    0 & \text{if , }A \ge 0, \\
    \pi & \text{if , }A \lt 0,
    \end{cases}##

    (I see mfb beat me to it !)
     
  5. Jan 4, 2014 #4
    Thanks guys for the help! : )
     
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