Santa Rosa, California is 7 miles due north of Rohnert Park. Bodega Bay is 19 miles due west of Rohnert Park (as the crow flies). What is the bearing of Santa Rosa from Bodega Bay?

Steps to Solve

1. Visualize the problem

Imagine a coordinate system where Rohnert Park is the origin (0,0). Santa Rosa is at (0,7) and Bodega Bay is at (-19, 0). We want to find the bearing of Santa Rosa from Bodega Bay, which means we want to find the angle formed at Bodega Bay between the north direction and the line connecting Bodega Bay to Santa Rosa.

2. Form a right triangle

We can form a right triangle with Bodega Bay, Rohnert Park, and Santa Rosa as the vertices. The legs of this triangle are 7 miles (north) and 19 miles (west).

3. Calculate the angle

Let $\theta$ be the angle at Bodega Bay between the west direction and the line connecting Bodega Bay to Santa Rosa. We can use the tangent function to find this angle:

$$ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{7}{19} $$

$$ \theta = \arctan\left(\frac{7}{19}\right) $$

Using a calculator, we get:

$$ \theta \approx 20.23^\circ $$

4. Determine the bearing

Since Bodega Bay is west of Rohnert Park, and Santa Rosa is north of Rohnert Park, Santa Rosa is to the north and east of Bodega Bay. The bearing is measured clockwise from the north. Since $\theta$ is the angle between the west direction and the line to Santa Rosa, we need to find the angle between the north direction and the line.

The angle between the North direction and Santa Rosa is $90^\circ - \theta$

$$ 90^\circ - 20.23^\circ \approx 69.77^\circ $$

Since the angle is measured clockwise from North, it implies the bearing is N69.77$^\circ$E

Another possible way to represent bearing is by starting from North (0 degrees) and going clockwise, so the bearing will be equal to $069.77^\circ$ T (T indicating True Bearing, since we didn't subtract any magnetic declination).

Rounding to the nearest degree, this is approximately $70^\circ$.