How does one find the rotation angle given a matrix?

Question

I have been having some trobule with this. I know that theta =     [costheta -sintheta]
                                                                                                       [sintheta costheta]
But I can't figure out how to find theta given a matrix, r      r  [-.5138772077.8578637511]                              
r                                                                                        [-.8578637511 -.5138772077] I thought that all I had to do was say costheta = -5.138772077 and solve for it. I solved for cos and sin and got values of 2.110494639 and -1.031098015. These numbers are incongruent and neither are apparently correct. I thought maybe I should find a value for tan but that too didn't work. Clearly I am messing up somewhere. Can someone explain how?

Hy · Accepted Answer

I don't know how you got two values for θ
Since cos and sin are both negative, we need the third quadrant value,
and so θ = π + arcsin(.8578637511)
 ............. = 4.172691 correct to six dec pl.

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How does one find the rotation angle given a matrix?

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