To find the measure of [tex]\(\theta\)[/tex] where [tex]\(0^\circ \leq \theta \leq 360^\circ\)[/tex] for [tex]\(\csc \theta = \sqrt{2}\)[/tex], follow these steps:
1. Understand the Relationship between [tex]\(\csc \theta\)[/tex] and [tex]\(\sin \theta\)[/tex]:
The cosecant function is the reciprocal of the sine function, so:
[tex]\[
\csc \theta = \frac{1}{\sin \theta}
\][/tex]
Given:
[tex]\[
\csc \theta = \sqrt{2}
\][/tex]
2. Find [tex]\(\sin \theta\)[/tex]:
Since [tex]\(\csc \theta = \sqrt{2}\)[/tex],
[tex]\[
\sin \theta = \frac{1}{\csc \theta} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}
\][/tex]
3. Determine [tex]\(\theta\)[/tex] using the arcsine function:
The principal value of [tex]\(\theta\)[/tex] (the solution in the first quadrant) can be found using the arcsine function:
[tex]\[
\theta = \arcsin\left(\frac{\sqrt{2}}{2}\right)
\][/tex]
The value of [tex]\(\theta\)[/tex] is:
[tex]\[
\theta_1 = 45^\circ
\][/tex]
4. Identify Additional Solutions:
Since the sine function is positive in both the first and second quadrants, we need to find:
- The solution in the second quadrant:
[tex]\[
\theta_2 = 180^\circ - \theta_1 = 180^\circ - 45^\circ = 135^\circ
\][/tex]
Additionally, considering the periodic nature of the sine function:
- The solution in the third quadrant:
[tex]\[
\theta_3 = 180^\circ + \theta_1 = 180^\circ + 45^\circ = 225^\circ
\][/tex]
- The solution in the fourth quadrant:
[tex]\[
\theta_4 = 360^\circ - \theta_1 = 360^\circ - 45^\circ = 315^\circ
\][/tex]
Therefore, the measures of [tex]\(\theta\)[/tex] that satisfy [tex]\(\csc \theta = \sqrt{2}\)[/tex] within the interval [tex]\(0^\circ \leq \theta \leq 360^\circ\)[/tex] are:
[tex]\[
\boxed{45.0^\circ, 135.0^\circ, 225.0^\circ, 315.0^\circ}
\][/tex]
