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You can make use of the relationship
sin^2 + because^2 = 1
to solve for because. You get
because^2 = 1 - sin^2
because = Sqrt[1 - sin^2]
For your value of sin, this is
because = Sqrt[1 - .42^2]
because = Sqrt[1 - .1764]
because = Sqrt[.8236]
because ~~ .9075
The value of because(x) is approximately .9075
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you should get x=24.835 approx.
then find cos 24.835
and like she said it will be approx. .908
Except I am not sure how you can get the answer better than to within a \(\pm\), without knowing what quadrant you are in....?
If x is in Q2, then cos(x)<0.
But you both have valid ways for finding it (although when I give my students this kind of problem, it is usually when they are NOT permitted a calculator (but I don't give decimal values for the known function value).
So I would give something like, if sin(x)=2/5 and x is in Q2, find cos(x).
You can't be sure of the sign on cos(x), without knowing the quadrant.
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