Navigating Trigonometry: Calculating the Closest Distance a Ship will be from a Lighthouse

When a ship is sailing and its path needs to be calculated relative to a lighthouse, a basic understanding of trigonometry is essential. In this article, we will explore a practical example where a ship is 30 km north of a lighthouse and is sailing on a bearing of 060°. We will calculate the closest distance the ship will be from the lighthouse. This scenario involves trigonometry, navigation, and the application of distance calculation techniques.

Positioning the Lighthouse and Ship

Let's start by positioning the lighthouse at the origin (0, 0). The ship is initially 30 km north of the lighthouse, so its starting position is at (0, 30).

Understanding the Bearing

A bearing of 060° means the ship is sailing 60° clockwise from true north. In standard Cartesian coordinates, this translates to an angle of 30° from the positive x-axis (east). The direction vector for the ship can be represented as:

Direction (cos 60°, sin 60°) (1/2, √3/2)

Finding the Closest Distance

The closest distance from the lighthouse to the ship's path will occur along a line that is perpendicular to the ship's direction. The line from the lighthouse to the point where the ship is closest can be found by using the angle of 90° to the bearing of 60°, which is 150°. The slope of the line perpendicular to the ship's direction can be calculated as follows:

The slope of the direction vector bearing 60° is tan 60° √3.

Therefore, the slope of the line perpendicular to it is -1/√3.

Setting Up the Equation

The equation of the line for the ship starting at (0, 30) with a slope of √3 is:

y - 30 √3x - 0 implies y √3x 30

The equation of the line from the lighthouse with a slope of -1/√3 is:

y -1/√3x

Finding the Intersection

To find the intersection point, set the two equations equal to each other:

√3x 30 -1/√3x

√3x 1/√3x -30

(√3 1/√3)x -30 implies x(3 1)/√3 -30 implies x -30 × (√3/4) -15√3/2

Calculating the Closest Distance

Substitute x back into either equation to find y:

y -1/√3(-15√3/2) 15/2

The intersection point is (-15√3/2, 15/2).

The distance from the lighthouse (0, 0) to this point is calculated as:

d √( (-15√3/2)^2 (15/2)^2 ) √( 675/4 225/4 ) √( 900/4 ) √225 15 km

Therefore, the closest distance the ship will be from the lighthouse is 15 km.

Keywords: trigonometry, navigation, ship distance calculation