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To solve this problem, we need to relate the distance sailed (245 nautical miles) to the change in longitude (7°25') and find the latitude at which the vessel is sailing.

### Step-by-step breakdown:
1. **Nautical miles and latitude relationship**:
   - At any latitude \( \phi \), the eastward distance a vessel covers corresponds to the change in longitude. The formula that relates the distance \( D \) sailed east to the change in longitude \( \Delta \lambda \) (in degrees) is given by:
   \[
   D = \Delta \lambda \cdot \cos(\phi) \cdot 60
   \]
   where:
   - \( D \) is the distance in nautical miles.
   - \( \Delta \lambda \) is the change in longitude in degrees (converted to decimal).
   - \( \phi \) is the latitude (in degrees).
   - The factor 60 arises because each degree of longitude corresponds to 60 nautical miles at the equator (latitude 0°).

2. **Convert the change in longitude to decimal**:
   - \( 7^\circ 25' \) can be converted to decimal form as:
   \[
   7^\circ 25' = 7 + \frac{25}{60} = 7.4167^\circ
   \]

3. **Rearrange the formula to solve for latitude**:
   - Rearranging the formula for latitude \( \phi \):
   \[
   \cos(\phi) = \frac{D}{\Delta \lambda \cdot 60}
   \]
   Substituting the known values:
   \[
   \cos(\phi) = \frac{245}{7.4167 \cdot 60} = \frac{245}{444.999}
   \]
   \[
   \cos(\phi) \approx 0.5506
   \]

4. **Find the latitude**:
   - Take the inverse cosine to find the latitude:
   \[
   \phi = \cos^{-1}(0.5506)
   \]
   Using a calculator:
   \[
   \phi \approx 56.58^\circ
   \]

### Final Answer:
The vessel was at a latitude of approximately **56.58°**.

Would you like further clarification or details on any part of this solution? Here are five related questions to explore:

1. How would the result change if the vessel sailed at a latitude of 30°?
2. What is the formula for calculating changes in longitude when sailing at different latitudes?
3. How does the curvature of the Earth affect distance calculations between longitudes?
4. Can this method be used for vessels sailing due west?
5. What would be the impact on the result if the distance sailed were measured in kilometers instead of nautical miles?

**Tip**: Always ensure that the longitude change is converted into decimal form before applying trigonometric calculations!

By sailing due east for a distance of 245 NM, a vessel changes its longitude by 7°25'. What latitude was the vessel in?

Calculate the latitude of a vessel sailing due east for 245 nautical miles, changing longitude by 7°25'. Understand how trigonometric principles and navigational formulas are applied to determine the vessel's position.

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