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I'd like to calculate an angle using nothing but either tangent, cosine, or sine, or any combination of the two. Is this possible? If so, how?

Why I'm asking this

I know how to program, but I find that I can't really get much farther until I start learning higher level mathematics. Therefore, I'm writing out C functions to teach me maths :D.

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    So this is for programming? You'll want to look into two-argument arctangent (`atan2()` in some languages).2011-12-24

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You want to learn programming and advanced mathematics then find root of $f(\phi) = \cos\theta - \cos \phi $

using Bisection method or Newton-Raphson method. Writing program for any of the methods is easy.

Note: Here $\cos \theta$ is value of cosine given to you and $\phi$ is your guess.

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Inverse Trigonometric Functions - Wikipedia

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    Understandable. I'll be sure to note that the next time I ask a question.2011-12-25