To determine the bearing from the cliff to the island, we need to consider the path taken by the puffin. The puffin flew 34 km due east and then 56 km due north to reach the island. To find the bearing, we need to follow these steps:
1. Identify the distances involved:
- Eastward distance = 34 km
- Northward distance = 56 km
2. Understand the relationships in the right-angled triangle formed by the path:
- The eastward distance forms the adjacent side
- The northward distance forms the opposite side
3. Calculate the angle to the bearing:
- To find the angle (θ) to the bearing, we use the tangent function in trigonometry, which is defined as tan(θ) = opposite / adjacent
[tex]\[
\tan(\theta) = \frac{\text{east distance}}{\text{north distance}} = \frac{34}{56}
\][/tex]
4. Find the angle [tex]\( \theta \)[/tex] using the arctangent (inverse tangent):
[tex]\[
\theta = \arctan\left(\frac{34}{56}\right)
\][/tex]
By calculating this angle, we get the angle θ in radians.
5. Convert the angle θ from radians to degrees:
[tex]\[
\theta_{\text{degrees}} \approx 31.26 \text{ degrees (using \(\arctan(34/56)\) and conversion to degrees)}
\][/tex]
6. Adjust the bearing:
- Since bearings are measured clockwise from due north:
[tex]\[
\text{Bearing from north} = \theta_{\text{degrees}}
\][/tex]
7. Ensure the bearing is within 0 to 360 degrees range:
- The computed bearing of approximately 31.26 degrees is already within this range.
Thus, to the nearest degree, the bearing from the cliff to the island is:
[tex]\[
\boxed{31 \text{ degrees}}
\][/tex]
This bearing means that the path from the cliff to the island is 31 degrees east of due north.
