Find the values of `theta` lying between `0^(@)` and `360^(@)` when
`sin theta=sin 21^(@)`

Answer

To find the values of \(\theta\) lying between \(0^\circ\) and \(360^\circ\) when \(\sin \theta = \sin 21^\circ\), we can use the properties of the sine function.

### Step-by-step Solution:

1. **Understanding the Sine Function**:
The sine function has a periodicity of \(360^\circ\) and is symmetric about \(180^\circ\). This means that if \(\sin \theta = \sin \alpha\), then:
\[ \theta = \alpha + 360n \quad \text{or} \quad \theta = 180 - \alpha + 360n \]
where \(n\) is any integer.

2. **Applying the General Solutions**:
For our case, \(\alpha = 21^\circ\). Therefore, we can write:
\[ \theta = 21^\circ + 360n \quad \text{or} \quad \theta = 180^\circ - 21^\circ + 360n \]

3. **Finding Specific Values**:
We need to find values of \(\theta\) for \(n = 0\) and \(n = 1\) that lie between \(0^\circ\) and \(360^\circ\).

- **For \(n = 0\)**:
- From \(\theta = 21^\circ + 360(0)\):
\[ \theta = 21^\circ \]
- From \(\theta = 180^\circ - 21^\circ + 360(0)\):
\[ \theta = 180^\circ - 21^\circ = 159^\circ \]

- **For \(n = 1\)**:
- From \(\theta = 21^\circ + 360(1)\):
\[ \theta = 21^\circ + 360 = 381^\circ \quad (\text{not valid since } 381^\circ > 360^\circ) \]
- From \(\theta = 180^\circ - 21^\circ + 360(1)\):
\[ \theta = 180^\circ - 21^\circ + 360 = 159^\circ + 360 = 519^\circ \quad (\text{not valid since } 519^\circ > 360^\circ) \]

4. **Conclusion**:
The valid values of \(\theta\) that lie between \(0^\circ\) and \(360^\circ\) are:
\[ \theta = 21^\circ \quad \text{and} \quad \theta = 159^\circ \]

### Final Answer:
The values of \(\theta\) are \(21^\circ\) and \(159^\circ\).

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