Answer :
To solve for [tex]\( \csc^{-1}(0.9) \)[/tex], we need to understand the nature of the cosecant function and its inverse.
1. Cosecant and its reciprocal: The cosecant function is the reciprocal of the sine function. Therefore, [tex]\( \csc^{-1}(y) \)[/tex] is equivalent to [tex]\( \sin^{-1}(1/y) \)[/tex]. Thus, we need to find [tex]\( \sin^{-1}(1/0.9) \)[/tex].
2. Calculate the reciprocal: Determine the value of [tex]\( 1/0.9 \)[/tex]:
[tex]\[ \frac{1}{0.9} \approx 1.1111 \][/tex]
3. Check the domain of the arcsine function: The arcsine function [tex]\( \sin^{-1}(x) \)[/tex] is defined only for [tex]\( x \)[/tex] such that [tex]\( -1 \leq x \leq 1 \)[/tex]. Here, [tex]\( 1.1111 \)[/tex] is outside this range. Therefore, the expression [tex]\( \sin^{-1}(1/0.9) \)[/tex] does not exist.
Since [tex]\( \sin^{-1}(1/0.9) \)[/tex] is not defined due to the fact that its input value [tex]\( 1.1111 \)[/tex] is outside the valid range for the arcsine function, the answer to [tex]\( \csc^{-1}(0.9) \)[/tex] does not exist (DNE).
1. Cosecant and its reciprocal: The cosecant function is the reciprocal of the sine function. Therefore, [tex]\( \csc^{-1}(y) \)[/tex] is equivalent to [tex]\( \sin^{-1}(1/y) \)[/tex]. Thus, we need to find [tex]\( \sin^{-1}(1/0.9) \)[/tex].
2. Calculate the reciprocal: Determine the value of [tex]\( 1/0.9 \)[/tex]:
[tex]\[ \frac{1}{0.9} \approx 1.1111 \][/tex]
3. Check the domain of the arcsine function: The arcsine function [tex]\( \sin^{-1}(x) \)[/tex] is defined only for [tex]\( x \)[/tex] such that [tex]\( -1 \leq x \leq 1 \)[/tex]. Here, [tex]\( 1.1111 \)[/tex] is outside this range. Therefore, the expression [tex]\( \sin^{-1}(1/0.9) \)[/tex] does not exist.
Since [tex]\( \sin^{-1}(1/0.9) \)[/tex] is not defined due to the fact that its input value [tex]\( 1.1111 \)[/tex] is outside the valid range for the arcsine function, the answer to [tex]\( \csc^{-1}(0.9) \)[/tex] does not exist (DNE).
