If `sin x^(@)=sinalpha , "then " alpha " is"`

Options

A. `(180)/(pi)`
B. `(pi)/(270)`
C. `(270)/(pi)`
D. `(pi)/(180)`

Explanation

To solve the equation \( \sin x^\circ = \sin \alpha \), we need to find the value of \( \alpha \) in terms of \( x \).

### Step-by-Step Solution:

1. **Understanding the Equation**:
We start with the equation:
\[ \sin x^\circ = \sin \alpha \]

2. **Using the Sine Function Property**:
The sine function has the property that if \( \sin A = \sin B \), then:
\[ A = B + 360k \quad \text{or} \quad A = 180 - B + 360k \]
where \( k \) is any integer.

3. **Applying the Property**:
From our equation, we can set up two cases:
- Case 1:
\[ x^\circ = \alpha + 360k \]
- Case 2:
\[ x^\circ = 180 - \alpha + 360k \]

4. **Solving for \( \alpha \)**:
- From Case 1:
\[ \alpha = x^\circ - 360k \]
- From Case 2:
\[ \alpha = 180 - x^\circ + 360k \]

5. **General Solution**:
Therefore, the general solutions for \( \alpha \) can be expressed as:
\[ \alpha = x^\circ + 360k \quad \text{or} \quad \alpha = 180 - x^\circ + 360k \]
where \( k \) is any integer.

6. **Considering the Range of \( \alpha \)**:
If we restrict \( \alpha \) to a specific range, for example, \( -90^\circ \leq \alpha \leq 90^\circ \), we need to consider only the relevant solutions from the above cases.

### Final Answer:
Thus, the values of \( \alpha \) can be expressed as:
\[ \alpha = x^\circ + 360k \quad \text{or} \quad \alpha = 180 - x^\circ + 360k \]
where \( k \in \mathbb{Z} \).