Answer :
To find the values of [tex]\( x \)[/tex] for which [tex]\( \sin x = -0.8176 \)[/tex] in the interval [tex]\( 0^{\circ} \leq x \leq 360^{\circ} \)[/tex], follow these steps:
1. Finding the Principal Value:
First, we determine the angle [tex]\( \theta \)[/tex] such that [tex]\( \sin(\theta) = -0.8176 \)[/tex]. This angle, known as the principal value, is typically found using the inverse sine function.
The principal value is approximately [tex]\( -54.84^{\circ} \)[/tex].
2. Adjusting to Find Angles in the Given Interval:
The sine function is negative in the third and fourth quadrants. We need to convert the principal value to angles that lie within these specific quadrants, while ensuring the angles fit within the interval [tex]\( 0^{\circ} \leq x \leq 360^{\circ} \)[/tex].
- For the third quadrant:
- A complete revolution (or [tex]\( 360^{\circ} \)[/tex]) subtracted by the magnitude of the principal angle gives the desired angle in the third quadrant:
[tex]\[ x1 = 180^{\circ} + (-54.84^{\circ}) = 180^{\circ} - 54.84^{\circ} \][/tex]
Simplifying this:
[tex]\[ x1 \approx 180^{\circ} - 54.84^{\circ} = 125.16^{\circ} \][/tex]
Rounding to the nearest degree, we get:
[tex]\[ x1 \approx 125^{\circ} \][/tex]
- For the fourth quadrant:
- A complete revolution (or [tex]\( 360^{\circ} \)[/tex]) minus the principal angle (considering its absolute value) gives a solution in the fourth quadrant:
[tex]\[ x2 = 360^{\circ} - (-54.84^{\circ}) = 360^{\circ} + 54.84^{\circ} \][/tex]
Simplifying this:
[tex]\[ x2 \approx 360^{\circ} + 54.84^{\circ} = 414.84^{\circ} \][/tex]
Since the interval is [tex]\( 0^{\circ} \leq x \leq 360^{\circ} \)[/tex], [tex]\( 414.84^{\circ} \)[/tex] is effectively equivalent to [tex]\( 414.84^{\circ} - 360^{\circ} \)[/tex]. Therefore:
[tex]\[ x2 \approx 54.84^{\circ} \][/tex]
Rounding to the nearest degree:
[tex]\[ x2 \approx 55^{\circ} \][/tex]
However, this value falls outside our adjusted answer, so rather we use another equivalent angle:
[tex]\[ 360^{\circ} - 54.84^{\circ} = 305.16^{\circ} \][/tex]
Rounding to the nearest degree:
[tex]\[ x2 \approx 415^{\circ} \][/tex]
3. Final Answer:
The angles in the interval [tex]\( 0^{\circ} \leq x \leq 360^{\circ} \)[/tex] for which [tex]\( \sin x = -0.8176 \)[/tex] are:
[tex]\[ x \approx 125^{\circ} \quad \text{and} \quad 415^{\circ} \][/tex]
Therefore, the values of [tex]\( x \)[/tex] for which [tex]\( \sin x = -0.8176 \)[/tex] in the specified interval are [tex]\( 125^{\circ} \)[/tex] and [tex]\( 415^{\circ} \)[/tex].
1. Finding the Principal Value:
First, we determine the angle [tex]\( \theta \)[/tex] such that [tex]\( \sin(\theta) = -0.8176 \)[/tex]. This angle, known as the principal value, is typically found using the inverse sine function.
The principal value is approximately [tex]\( -54.84^{\circ} \)[/tex].
2. Adjusting to Find Angles in the Given Interval:
The sine function is negative in the third and fourth quadrants. We need to convert the principal value to angles that lie within these specific quadrants, while ensuring the angles fit within the interval [tex]\( 0^{\circ} \leq x \leq 360^{\circ} \)[/tex].
- For the third quadrant:
- A complete revolution (or [tex]\( 360^{\circ} \)[/tex]) subtracted by the magnitude of the principal angle gives the desired angle in the third quadrant:
[tex]\[ x1 = 180^{\circ} + (-54.84^{\circ}) = 180^{\circ} - 54.84^{\circ} \][/tex]
Simplifying this:
[tex]\[ x1 \approx 180^{\circ} - 54.84^{\circ} = 125.16^{\circ} \][/tex]
Rounding to the nearest degree, we get:
[tex]\[ x1 \approx 125^{\circ} \][/tex]
- For the fourth quadrant:
- A complete revolution (or [tex]\( 360^{\circ} \)[/tex]) minus the principal angle (considering its absolute value) gives a solution in the fourth quadrant:
[tex]\[ x2 = 360^{\circ} - (-54.84^{\circ}) = 360^{\circ} + 54.84^{\circ} \][/tex]
Simplifying this:
[tex]\[ x2 \approx 360^{\circ} + 54.84^{\circ} = 414.84^{\circ} \][/tex]
Since the interval is [tex]\( 0^{\circ} \leq x \leq 360^{\circ} \)[/tex], [tex]\( 414.84^{\circ} \)[/tex] is effectively equivalent to [tex]\( 414.84^{\circ} - 360^{\circ} \)[/tex]. Therefore:
[tex]\[ x2 \approx 54.84^{\circ} \][/tex]
Rounding to the nearest degree:
[tex]\[ x2 \approx 55^{\circ} \][/tex]
However, this value falls outside our adjusted answer, so rather we use another equivalent angle:
[tex]\[ 360^{\circ} - 54.84^{\circ} = 305.16^{\circ} \][/tex]
Rounding to the nearest degree:
[tex]\[ x2 \approx 415^{\circ} \][/tex]
3. Final Answer:
The angles in the interval [tex]\( 0^{\circ} \leq x \leq 360^{\circ} \)[/tex] for which [tex]\( \sin x = -0.8176 \)[/tex] are:
[tex]\[ x \approx 125^{\circ} \quad \text{and} \quad 415^{\circ} \][/tex]
Therefore, the values of [tex]\( x \)[/tex] for which [tex]\( \sin x = -0.8176 \)[/tex] in the specified interval are [tex]\( 125^{\circ} \)[/tex] and [tex]\( 415^{\circ} \)[/tex].
