To find a value of [tex]\( s \)[/tex] in the interval [tex]\([0, \frac{\pi}{2}]\)[/tex] that satisfies [tex]\(\csc s = 1.4719\)[/tex], follow these steps:
1. Understanding the cosecant function:
The cosecant function is the reciprocal of the sine function. Therefore, given [tex]\(\csc s = 1.4719\)[/tex], it implies:
[tex]\[
\sin s = \frac{1}{\csc s}
\][/tex]
2. Calculate [tex]\(\sin s\)[/tex]:
Substitute the given value of [tex]\(\csc s\)[/tex]:
[tex]\[
\sin s = \frac{1}{1.4719}
\][/tex]
Compute this value:
[tex]\[
\sin s \approx 0.6796
\][/tex]
3. Determine [tex]\( s \)[/tex] from [tex]\(\sin s\)[/tex]:
We need to find the angle [tex]\( s \)[/tex] in the interval [tex]\([0, \frac{\pi}{2}]\)[/tex] whose sine is approximately 0.6796. To do this, we use the inverse sine function (arcsine), which gives us an angle for a given sine value:
[tex]\[
s = \arcsin(0.6796)
\][/tex]
Calculate this value:
[tex]\[
s \approx 0.746936424347283
\][/tex]
4. Verify the interval:
We need to ensure that the computed value of [tex]\( s \)[/tex] lies within the specified interval [tex]\([0, \frac{\pi}{2}]\)[/tex]. Since:
[tex]\[
0 \leq 0.746936424347283 \leq \frac{\pi}{2} \approx 1.5708
\][/tex]
The value of [tex]\( s \)[/tex] lies within the given interval.
Thus, the value of [tex]\( s \)[/tex] that satisfies [tex]\(\csc s = 1.4719\)[/tex] in the interval [tex]\([0, \frac{\pi}{2}]\)[/tex] is approximately:
[tex]\[
s \approx 0.746936424347283
\][/tex]
