To find the value of [tex]\(\theta\)[/tex] in the interval [tex]\(0^\circ \leq \theta \leq 90^\circ\)[/tex] that satisfies the equation [tex]\(\sin \theta = 0.68375982\)[/tex], we need to determine [tex]\(\theta\)[/tex] such that:
[tex]\[
\theta = \sin^{-1}(0.68375982)
\][/tex]
The inverse sine function, also known as arcsine, will give us the angle whose sine is the given value. We use a calculator to determine this angle in degrees.
Given the value [tex]\(\sin \theta = 0.68375982\)[/tex]:
[tex]\[
\theta = \sin^{-1}(0.68375982) \approx 43.138151907853036^\circ
\][/tex]
To provide a simplified answer, we round the result to six decimal places:
[tex]\[
\theta \approx 43.138152^\circ
\][/tex]
Therefore, the value of [tex]\(\theta\)[/tex] that satisfies [tex]\(\sin \theta = 0.68375982\)[/tex] within the interval [tex]\(0^\circ \leq \theta \leq 90^\circ\)[/tex] is:
[tex]\[
\theta \approx 43.138152^\circ
\][/tex]
