State a domain for cos theta that would make its inverse cos -1thete a functio n
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Graph the cosine function. You know its inverse is reflected about the line y = x. After you make this reflection, you can see that because cos θ is periodic, the reflected graph does not pass the vertical line test, and therefore is not a function. If, however, we restrict the domain of cos θ to half a period, then reflect it about y = x, the problem is eliminated.
So you can choose any range that is π radians (or 180°), but we use [0, π] because the standard angles lie in this range, and more importantly, it allows the arccos function to return a real answer for every value in the range of cos θ [-1,1].
Graph both functions with a graphing utility and see what I mean. That's really the best way to learn how the inverse trigonometric functions work.
Good luck!
So you can choose any range that is π radians (or 180°), but we use [0, π] because the standard angles lie in this range, and more importantly, it allows the arccos function to return a real answer for every value in the range of cos θ [-1,1].
Graph both functions with a graphing utility and see what I mean. That's really the best way to learn how the inverse trigonometric functions work.
Good luck!
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0 ≤ θ ≤ π
This is in fact the standard domain for arccos(x).
This is in fact the standard domain for arccos(x).