To solve the equation [tex]\(\sin x = 0\)[/tex] within the range [tex]\(0^{\circ} \leq x \leq 360^{\circ}\)[/tex], let's analyze the behavior of the sine function.
### Step-by-Step Solution:
1. Understand the Sine Function:
- The sine function, [tex]\(\sin x\)[/tex], describes the y-coordinate of a point on the unit circle as it moves counterclockwise from the positive x-axis.
- The sine function repeats its values in a periodic manner every [tex]\(360^{\circ}\)[/tex]. This is called the period of the sine function.
2. Identify the Key Points:
- The sine function equals zero where the y-coordinate of the corresponding point on the unit circle is zero.
- These points correspond to angles where the point lies on the x-axis (since the y-coordinate is 0 on the x-axis).
3. Determine the Exact Angles:
- Starting at [tex]\(x = 0^{\circ}\)[/tex], the point is at [tex]\((1, 0)\)[/tex] on the unit circle, so [tex]\(\sin 0^{\circ} = 0\)[/tex].
- As the angle continues around the circle, another point where the sine function equals zero is [tex]\(x = 180^{\circ}\)[/tex], where the point is at [tex]\((-1, 0)\)[/tex].
- Finally, completing the full circle, [tex]\(\sin x\)[/tex] returns to zero at [tex]\(x = 360^{\circ}\)[/tex].
4. Compile the Solutions:
- Therefore, the angles [tex]\(x\)[/tex] for which [tex]\(\sin x = 0\)[/tex] and within the given range [tex]\(0^{\circ} \leq x \leq 360^{\circ}\)[/tex] are:
- [tex]\(x = 0^{\circ}\)[/tex]
- [tex]\(x = 180^{\circ}\)[/tex]
- [tex]\(x = 360^{\circ}\)[/tex]
Hence, the solutions to the equation [tex]\(\sin x = 0\)[/tex] within the specified range are:
[tex]\[
x = 0^{\circ}, \quad x = 180^{\circ}, \quad \text{and} \quad x = 360^{\circ}.
\][/tex]
