To find the values of [tex]\( x \)[/tex] in the interval [tex]\( 0^\circ \leq x \leq 360^\circ \)[/tex] for which [tex]\(\sin x = -0.8176\)[/tex], follow these steps:
1. Understand the given value:
We are given that [tex]\(\sin x = -0.8176\)[/tex]. The sine function is negative in the third and fourth quadrants.
2. Find the reference angle:
To find the reference angle, we take the inverse sine (arcsin) of the absolute value of the given sine value. The reference angle is calculated as:
[tex]\[
\text{ref\_angle} = \sin^{-1}(0.8176)
\][/tex]
This results in:
[tex]\[
\text{ref\_angle} \approx 54.85^\circ
\][/tex]
3. Determine the angles in the third and fourth quadrants:
Since [tex]\(\sin x\)[/tex] is negative, [tex]\( x \)[/tex] can be in the third quadrant (between [tex]\( 180^\circ \)[/tex] and [tex]\( 270^\circ \)[/tex]) or in the fourth quadrant (between [tex]\( 270^\circ \)[/tex] and [tex]\( 360^\circ \)[/tex]).
- For the third quadrant:
The angle [tex]\( x \)[/tex] in the third quadrant is given by:
[tex]\[
x_1 = 180^\circ + \text{ref\_angle}
\][/tex]
Substituting the reference angle:
[tex]\[
x_1 = 180^\circ + 54.85^\circ \approx 234.85^\circ
\][/tex]
Rounded to the nearest degree:
[tex]\[
x_1 \approx 235^\circ
\][/tex]
- For the fourth quadrant:
The angle [tex]\( x \)[/tex] in the fourth quadrant is given by:
[tex]\[
x_2 = 360^\circ - \text{ref\_angle}
\][/tex]
Substituting the reference angle:
[tex]\[
x_2 = 360^\circ - 54.85^\circ \approx 305.15^\circ
\][/tex]
Rounded to the nearest degree:
[tex]\[
x_2 \approx 305^\circ
\][/tex]
In summary, the values of [tex]\( x \)[/tex] in the interval [tex]\( 0^\circ \leq x \leq 360^\circ \)[/tex] for which [tex]\(\sin x = -0.8176\)[/tex] are:
[tex]\[
x \approx 235^\circ \quad \text{and} \quad x \approx 305^\circ
\][/tex]
These are the answers to the nearest degree.
