Understanding Bearings and Navigation: Calculating the Three-Figure Bearing of Point C from A

In navigation and navigation-related fields, understanding bearings and how to calculate them is essential. This article will guide you through the process of finding the three-figure bearing of point C from A, given the coordinates and bearings of points B and C relative to point A. We will use trigonometric concepts and spherical trigonometry to solve this problem step by step.

Step-by-Step Calculation

1. Understanding the Problem

Let's assume the following information is given:

Point B is 10 km on a bearing of 030° from point A. Point C is 10 km on a bearing of 180° from point B.

Our goal is to find the three-figure bearing of point C from point A.

2. Placing Point A

We can consider point A to be at the origin (0, 0).

3. Finding Coordinates of Point B

- Point B is located 10 km on a bearing of 030° from point A.

- The bearing of 030° means it is 30° clockwise from north.

- To find the coordinates of B:

Using trigonometric functions:

B_x 10 times sin(30°) 10 times 0.5 5 B_y 10 times cos(30°) 10 times frac{sqrt{3}}{2} approx 8.66

The coordinates of point B are approximately 5 8.66.

4. Finding Coordinates of Point C

- Point C is located 10 km on a bearing of 180° from point B.

- A bearing of 180° is directly south.

- To find the coordinates of C:

Since C is 10 km directly south of B, we subtract 10 from the y-coordinate of B.

C_x B_x 5 C_y B_y - 10 8.66 - 10 -1.34

The coordinates of point C are approximately 5 -1.34.

5. Calculating the Bearing of C from A

- The coordinates of A are 0 0.

- The coordinates of C are 5 -1.34.

To find the angle from A to C, we use the tangent function:

begin{align*} tan(theta) frac{C_y - A_y}{C_x - A_x} frac{-1.34 - 0}{5 - 0} frac{-1.34}{5} theta text{atan2}(-1.34, 5) end{align*}

Calculating this gives us an angle of approximately -15.1° from the positive x-axis east.

To convert this to a bearing clockwise from north:

The angle from north is: 90° - (-15.1°) 90° 15.1° 105.1°.

6. Final Answer

Rounding to the nearest whole number, the three-figure bearing of C from A is 105°.

Thus, the three-figure bearing of C from A is 105°.

Another Approach: Using Spherical Trigonometry

Alternately, we can use a geometric approach based on the fact that Δ BAC is an isosceles triangle.

The angle ABC is equal to the difference between the bearing of A from B 30° and the bearing of C from B 180°, which is 210°. Hence, ABC 210° - 180° 30°. Since Δ ABC is isosceles with AB BC, angle CAB angle CBA 30°. The angle BAC is therefore 180° - 2 times 30° 120°. The bearing of C from A is equal to the sum of the bearing of B from A (30°) and the angle CAB (75°), which is 30° 75° 105°.

Thus, the three-figure bearing of C from A is 105°.

Conclusion

This article demonstrates two methods to find the three-figure bearing of point C from A. Whether using trigonometry or spherical trigonometry, both methods yield the same result. Understanding these calculations is crucial for various applications in navigation, surveying, and mathematics.