Answer :
To find all solutions of the equation \(2 \sin \theta - 1 = 0\) in the interval \([0, 2\pi)\), follow these detailed steps:
1. Isolate the Sine Function:
Start by isolating \(\sin \theta\) in the equation:
[tex]\[ 2 \sin \theta - 1 = 0 \][/tex]
Add 1 to both sides:
[tex]\[ 2 \sin \theta = 1 \][/tex]
Divide both sides by 2:
[tex]\[ \sin \theta = \frac{1}{2} \][/tex]
2. Determine the Reference Angle:
Next, we need to find the angle \(\theta\) within the given interval where \(\sin \theta = \frac{1}{2}\). Recall that the sine of an angle is \(\frac{1}{2}\) at:
[tex]\[ \theta = \frac{\pi}{6} \][/tex]
This angle is a well-known angle in the unit circle where the sine value is \(\frac{1}{2}\).
3. Find All Angles in the Interval \([0, 2\pi)\):
Since the sine function is positive in both the first and second quadrants, we need to find the corresponding angles in these quadrants.
- First Quadrant:
[tex]\[ \theta = \frac{\pi}{6} \][/tex]
- Second Quadrant:
The reference angle for the second quadrant is \( \pi - \frac{\pi}{6} \):
[tex]\[ \theta = \pi - \frac{\pi}{6} = \frac{6\pi}{6} - \frac{\pi}{6} = \frac{5\pi}{6} \][/tex]
4. State the Final Solutions:
The angles \(\theta\) within the interval \([0, 2\pi)\) that satisfy the equation \(2 \sin \theta - 1 = 0\) are:
[tex]\[ \theta = \frac{\pi}{6}, \frac{5\pi}{6} \][/tex]
Thus, the final solutions are:
[tex]\[ \boxed{\frac{\pi}{6}, \frac{5\pi}{6}} \][/tex]
Therefore, the solutions to the equation [tex]\(2 \sin \theta - 1 = 0\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] are [tex]\( \theta = \frac{\pi}{6} \)[/tex] and [tex]\( \theta = \frac{5\pi}{6} \)[/tex].
1. Isolate the Sine Function:
Start by isolating \(\sin \theta\) in the equation:
[tex]\[ 2 \sin \theta - 1 = 0 \][/tex]
Add 1 to both sides:
[tex]\[ 2 \sin \theta = 1 \][/tex]
Divide both sides by 2:
[tex]\[ \sin \theta = \frac{1}{2} \][/tex]
2. Determine the Reference Angle:
Next, we need to find the angle \(\theta\) within the given interval where \(\sin \theta = \frac{1}{2}\). Recall that the sine of an angle is \(\frac{1}{2}\) at:
[tex]\[ \theta = \frac{\pi}{6} \][/tex]
This angle is a well-known angle in the unit circle where the sine value is \(\frac{1}{2}\).
3. Find All Angles in the Interval \([0, 2\pi)\):
Since the sine function is positive in both the first and second quadrants, we need to find the corresponding angles in these quadrants.
- First Quadrant:
[tex]\[ \theta = \frac{\pi}{6} \][/tex]
- Second Quadrant:
The reference angle for the second quadrant is \( \pi - \frac{\pi}{6} \):
[tex]\[ \theta = \pi - \frac{\pi}{6} = \frac{6\pi}{6} - \frac{\pi}{6} = \frac{5\pi}{6} \][/tex]
4. State the Final Solutions:
The angles \(\theta\) within the interval \([0, 2\pi)\) that satisfy the equation \(2 \sin \theta - 1 = 0\) are:
[tex]\[ \theta = \frac{\pi}{6}, \frac{5\pi}{6} \][/tex]
Thus, the final solutions are:
[tex]\[ \boxed{\frac{\pi}{6}, \frac{5\pi}{6}} \][/tex]
Therefore, the solutions to the equation [tex]\(2 \sin \theta - 1 = 0\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] are [tex]\( \theta = \frac{\pi}{6} \)[/tex] and [tex]\( \theta = \frac{5\pi}{6} \)[/tex].
