Sure, let's solve the equation [tex]\(\sin(\theta) = \frac{2}{7}\)[/tex] for angles [tex]\(\theta\)[/tex] between [tex]\(0^{\circ}\)[/tex] and [tex]\(180^{\circ}\)[/tex].
1. Identify the value of sin([tex]\(\theta\)[/tex]):
We know that the sine of the angle [tex]\(\theta\)[/tex] is given by:
[tex]\[
\sin(\theta) = \frac{2}{7}
\][/tex]
2. Find the first angle [tex]\(\theta_1\)[/tex]:
To find [tex]\(\theta_1\)[/tex], we need to determine the angle whose sine is [tex]\(\frac{2}{7}\)[/tex]. The function [tex]\(\sin^{-1}(x)\)[/tex] (the inverse sine function, also called arcsine) will give us the principal angle:
[tex]\[
\theta_1 = \sin^{-1}\left(\frac{2}{7}\right)
\][/tex]
When we compute this, we get:
[tex]\[
\theta_1 \approx 16.6015^\circ
\][/tex]
3. Find the second angle [tex]\(\theta_2\)[/tex]:
Since the sine function is positive in both the first and second quadrants, there is another angle [tex]\(\theta_2\)[/tex] in the second quadrant that has the same sine value. This angle is given by:
[tex]\[
\theta_2 = 180^\circ - \theta_1
\][/tex]
Substituting [tex]\(\theta_1\)[/tex]:
[tex]\[
\theta_2 = 180^\circ - 16.6015^\circ = 163.3985^\circ
\][/tex]
4. Conclusion:
The angles [tex]\(\theta\)[/tex] between [tex]\(0^\circ\)[/tex] and [tex]\(180^\circ\)[/tex] that satisfy [tex]\(\sin(\theta) = \frac{2}{7}\)[/tex] are:
[tex]\[
\boxed{16.6015^\circ \text{ and } 163.3985^\circ}
\][/tex]
These are the detailed steps to find the angles [tex]\(\theta\)[/tex] such that [tex]\(\sin(\theta) = \frac{2}{7}\)[/tex] within the given range.
