If `cosx^(@)=sin 200^(@)`, find the possible values of x between `-180^(@)` and `360^(@)`.

Answer

To solve the equation \( \cos x = \sin 200^\circ \) and find the possible values of \( x \) between \( -180^\circ \) and \( 360^\circ \), we can follow these steps:

### Step-by-Step Solution:

1. **Use the Co-Function Identity**:
We know that \( \sin \theta = \cos(90^\circ - \theta) \). Therefore, we can rewrite \( \sin 200^\circ \) as:
\[ \sin 200^\circ = \cos(90^\circ - 200^\circ) = \cos(-110^\circ) \]
So, we have:
\[ \cos x = \cos(-110^\circ) \]

2. **Set Up the Cosine Equation**:
The equation \( \cos x = \cos(-110^\circ) \) implies:
\[ x = -110^\circ + 360^\circ n \quad \text{or} \quad x = 110^\circ + 360^\circ n \]
where \( n \) is any integer.

3. **Find Values in the Specified Range**:
We need to find values of \( x \) that lie between \( -180^\circ \) and \( 360^\circ \).

- For \( x = -110^\circ + 360^\circ n \):
- If \( n = 0 \): \( x = -110^\circ \) (valid)
- If \( n = 1 \): \( x = 250^\circ \) (valid)
- If \( n = -1 \): \( x = -470^\circ \) (not valid)

- For \( x = 110^\circ + 360^\circ n \):
- If \( n = 0 \): \( x = 110^\circ \) (valid)
- If \( n = 1 \): \( x = 470^\circ \) (not valid)
- If \( n = -1 \): \( x = -250^\circ \) (not valid)

4. **List the Valid Solutions**:
The valid solutions for \( x \) that lie between \( -180^\circ \) and \( 360^\circ \) are:
\[ x = -110^\circ, \quad x = 110^\circ, \quad x = 250^\circ \]

### Final Answer:
The possible values of \( x \) are:
\[ x = -110^\circ, \quad x = 110^\circ, \quad x = 250^\circ \]