Find the Domain csc(arccos(x))

Problem

csc(arccos(x))

Solution

1. Identify the inner function, which is y=arccos(x) The domain of the inverse cosine function is restricted to the interval [−1,1]

2. Determine the constraints for the outer function, csc(u)=1/sin(u) The cosecant function is undefined when its argument u results in sin(u)=0

3. Solve for the values of x where the expression is undefined by setting the denominator to zero: sin(arccos(x))=0

4. Apply the property that sin(arccos(x))=√(,1−x2) Setting this equal to zero gives √(,1−x2)=0 which implies 1−x2=0 or x=1 and x=−1

5. Exclude these boundary points from the initial domain of [−1,1] because they cause division by zero in the cosecant function.

6. Combine the restrictions to find the final interval where the expression is defined.

Final Answer

Domain of *csc(arccos(x))=(−1,1)

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