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Math Help - Finding missing angles of a right triangle without inverse operators

  1. #1
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    Finding missing angles of a right triangle without inverse operators

    I'm working on a little project with a basic programming language and I would like to find the missing angles of a right triangle but I don't have access to ArcSin, ArcCos, ArcTan.

    I do know the lengths of all legs of the triangle and I do have access to some math functions such as sin, cos, tan, sqrt, pi, and powers.

    So, given access to these functions but not to the inverse trig functions, is there a way I can use the information that I have to calculate the remaining angles in degrees?
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    Re: Finding missing angles of a right triangle without inverse operators

    You should use the Taylor series for these functions.
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    Re: Finding missing angles of a right triangle without inverse operators

    Given q, find \theta \in (0, \pi) such that \cos(\theta) = q:

    1. Newton's Method: f(\theta) = \cos(\theta) - q, so find \theta such that find f(\theta) = 0.

    2. \theta = \cos^{-1}(q) = \int_q^1 \frac{1}{\sqrt{1-x^2}} dx. Then use Trapezoid or Simpson's Method to do the integral.

    3. Power series for \theta = \cos^{-1}(q), based at a nearby point where you know the answer.

    4. Zero in on it: If q = 0.25, then \cos(1) = .54, \cos(1.5) = .07, so \theta between 1 and 1.5. Try 1.25:

    \cos(1.25) = .31, so \theta between 1.25 and 1.5. Try 1.37:

    \cos(1.37) = .20, so \theta between 1.25 and 1.37. Try 1.31:

    \cos(1.31) = .258, so \theta between 1.31 and 1.37. Etc...

    5. Depending on how many decimal points you want, you could build a table of cosine values, and then look it up (binary search). You could then do a linear interpolation between the successive table entries (or better, see #6).

    6. If you have a good guess at a point where you know the inverse trig function's value, you can estimate it's value at a nearby point by using the linear approximation to the inverse-trig function at that point. This could be combined with several of the other approaches. Moreover, yo could make this a quadartic approximation, or a cubic approximation, which will give an even better approximation for small changes.
    Last edited by johnsomeone; October 4th 2012 at 05:51 AM.
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