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Math Help - Trying to solve a Trig Equation

  1. #1
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    Trying to solve a Trig Equation

    Hi There,

    I've just joined the forum, and this is my first post. I'm an engineer working on problem with some angle offsets in a device, and would like to be able to solve the following equation. a, b and c are known variables, I am trying to solve for SF, is this possible?

    c = Sin-1(a/SF) - Sin-1(b/SF)
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  2. #2
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  3. #3
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    Re: Trying to solve a Trig Equation

    The result from WA looks horrendous, maybe it simplifies ? However, try this instead.

    First note that \sin^{-1}(a/SF) = \cos^{-1}(\sqrt{SF^2 - a^2}/SF), and similarly for \sin^{-1}(b/SF).

    Take the sine of the equation and making use of the expansion for \sin(A - B), arrive at

    SF^{2}\sin(c) = a\sqrt{SF^{2}-b^{2}}-b\sqrt{SF^{2}-a^{2}}\qquad \dots (1).

    Next, take the cosine of the original equation and having used the expansion for \cos(A - B), arrive at

    SF^{2}\cos(c) = \sqrt{SF^{2}-a^{2}}.\sqrt{SF^{2}-b^{2}}+ab \qquad \dots (2).

    Finally, square (1), make the obvious substitution for the product of the two square roots using (2) and then simplify to arrive at

    SF^2 = \frac{a^{2}+b^{2}-2ab\cos(c)}{\sin^{2}(c)}.
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