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To solve this problem, we need to calculate the bearing from the cliff to the island.

### Step 1: Understand the problem
- The puffin flies **38 km due east**, then **54 km due north**.
- The problem is to calculate the **bearing** from the cliff to the island.

Bearings are measured clockwise from the north. So, we are looking for the angle from the north, which we can find using trigonometry.

### Step 2: Visualize the triangle
We can form a right-angled triangle:
- One leg represents the **38 km** distance east.
- The other leg represents the **54 km** distance north.
- The angle we are looking for is the angle between the line going straight north (i.e., the vertical leg of the triangle) and the hypotenuse of this triangle (the straight line from the cliff to the island).

### Step 3: Use trigonometry
We can calculate the angle \( \theta \) (the angle from the east direction) using the **tangent** function:
\[
\tan(\theta) = \frac{\text{opposite (north direction)}}{\text{adjacent (east direction)}} = \frac{54}{38}
\]
Then, the angle \( \theta \) is:
\[
\theta = \tan^{-1} \left( \frac{54}{38} \right)
\]

Let me calculate that.The angle \( \theta \), measured from the eastward direction, is approximately **54.87°**.

### Step 4: Calculate the bearing
Since bearings are measured clockwise from the north, the bearing is calculated as:
\[
\text{Bearing} = 90^\circ - \theta
\]
Now, I'll compute the bearing.The bearing from the cliff to the island is approximately **35°**, measured clockwise from the north.

Would you like further details or have any other questions?

Here are some related questions:
1. How do you calculate the distance between two points using trigonometry?
2. What is the difference between bearings and compass directions?
3. How can we find the total distance traveled by the puffin?
4. How do we convert between radians and degrees in trigonometric calculations?
5. Can bearings be negative or exceed 360°?

**Tip:** Bearings are always measured clockwise from the north and are typically given in three-digit format, for example, 035°.

A puffin leaves a cliff and flies 38 km due east then 54 km due north to an island. Work out the bearing from the cliff to the island. Give your answer to the nearest degree.

Learn how to calculate the bearing from a cliff to an island after a puffin flies 38 km due east and 54 km due north. This problem involves right-angle triangle trigonometry and bearings, using the tangent function and Pythagorean theorem. Ideal for students in Grades 9-11.

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