To find the angle [tex]\(\theta\)[/tex] given that [tex]\(\sin \theta = \frac{11}{17}\)[/tex] and [tex]\(90^\circ < \theta < 180^\circ\)[/tex], follow these steps:
1. Find the sine of the angle:
[tex]\[
\sin \theta = \frac{11}{17} \approx 0.6471
\][/tex]
2. Calculate the reference angle:
Since [tex]\(90^\circ < \theta < 180^\circ\)[/tex], [tex]\(\theta\)[/tex] is in the second quadrant where [tex]\(\sin\)[/tex] is positive. The reference angle can be found using the inverse sine function (arcsine):
[tex]\[
\theta_{\text{ref}} = \sin^{-1}(0.6471)
\][/tex]
Without a calculator, this step would typically be done using tables or inverse function properties, but for this detailed solution, let's directly state the value:
[tex]\[
\theta_{\text{ref}} \approx 40.3^\circ
\][/tex]
3. Adjust for the second quadrant:
Since the angle is in the second quadrant, we have:
[tex]\[
\theta = 180^\circ - \theta_{\text{ref}}
\][/tex]
Given [tex]\(\theta_{\text{ref}} \approx 40.3^\circ\)[/tex],
[tex]\[
\theta \approx 180^\circ - 40.3^\circ = 139.7^\circ
\][/tex]
4. Convert the degree measure to radians (at least 4 decimal places):
We know [tex]\(180^\circ\)[/tex] corresponds to [tex]\(\pi\)[/tex] radians. So, to convert [tex]\(139.7^\circ\)[/tex] to radians:
[tex]\[
\theta_{\text{radians}} = 139.7^\circ \times \frac{\pi}{180^\circ}
\][/tex]
Plugging in the numbers:
[tex]\[
\theta_{\text{radians}} \approx 139.7 \times \frac{\pi}{180} \approx 2.4379 \, \text{radians}
\][/tex]
To summarize, the angle [tex]\(\theta\)[/tex]:
- In degree measure (to 1 decimal place): [tex]\(\theta \approx 139.7^\circ\)[/tex]
- In radian measure (to at least 4 decimal places): [tex]\(\theta \approx 2.4379\)[/tex] radians
