Problem 2 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

From the top of a light house 60 meters high with its base at sea level, the angle of depression is \(15^{\circ}\). The distance of the boat from the foot of the light house is (a) \(60 \times\left(\frac{\sqrt{3}-1}{\sqrt{3}+1}\right) m\) (b) \(60 \times\left(\frac{\sqrt{3}+1}{\sqrt{3}-1}\right) m\) (c) \(\left(\frac{\sqrt{3}+1}{\sqrt{3}-1}\right) m\) (d) None

Step-by-Step Solution

Answer

The distance of the boat from the foot of the lighthouse is \(60 \times\left(\frac{\sqrt{3}+1}{\sqrt{3}-1}\right) m\). So, option (b) is the correct answer.

Step 1: Understand the problem

A person is standing at the top of a 60m high lighthouse and looking at a boat in the sea. The line of sight to the boat forms an angle of depression of \(15^{\circ}\) with the horizontal line from the observer point. The task is to find the distance of the boat from the foot of the lighthouse. In other words, we should find the base of the triangle.

Step 2: Applying trigonometry

The concept of contemporarily using tangent of an angle comes into play here. In a right-angled triangle, the tangent of an angle is the ratio of the side opposite to it and the side adjacent to it. Therefore, we can write: \(\tan(15^{\circ}) = \frac{Height of lighthouse}{distance of boat from the foot of the lighthouse(our requirement)}\).

Step 3: Solving the equation

Cross multiplying, we get: Distance of the boat from the foot of the lighthouse \(= \frac{Height of lighthouse}{\tan(15^{\circ})} = \frac{60}{\tan(15^{\circ})} = 60 \times\left(\frac{\sqrt{3}+1}{\sqrt{3}-1}\right) m\). We have used the trigonometric value \(\tan(15^{\circ}) = \frac{\sqrt{3}-1}{\sqrt{3}+1}\).

Key Concepts

Trigonometric Ratios

Understanding trigonometric ratios is essential for solving problems involving right-angled triangles. In trigonometry, the primary ratios are sine, cosine, and tangent, which relate the angles of a triangle to the lengths of its sides. In a right-angled triangle:

• The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
• The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.
• The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.

These ratios can be remembered using the mnemonic 'SOH CAH TOA'.

These ratios are fundamental in solving many geometrical problems, including finding distances that are not easily measured, which is often the case in real-world scenarios. They enable us to work with the angles and sides of triangles effectively, particularly when we only know some of the information about the triangle and need to find out the rest.

Tangent in Right-Angled Triangles

The tangent ratio is particularly useful for right-angled triangles, where the angle in question is neither the 90-degree angle nor the right angle. It directly relates the angle's opposite side to the adjacent side. To use the tangent function in a right-angled triangle, remember that:

• The angle of interest is always the one that is not the right angle (90 degrees).
• The opposite side is the side that does not touch the angle of interest except at its endpoint.
• The adjacent side is the side that forms both the right angle and the angle of interest.

For example, if we have an angle of depression from a lighthouse top like the exercise provided, we can visualize a horizontal line (representing eye level) extending from the top of the lighthouse. The angle of depression is formed between this horizontal line and the line of sight to an object below. By considering this angle, and the vertical height of the lighthouse, we form a right-angled triangle of which we can use the tangent ratio to find the distance from the base to the object.

Trigonometry in Real-World Problems

Trigonometry isn't just for academic exercises; it has numerous practical applications in various fields like architecture, engineering, astronomy, physics, and even in casual navigation. Whenever we need to find a distance that we can't measure directly, trigonometry provides a toolset for indirect measurement using angles. For instance, an architect might use trigonometry to determine the height of a proposed building, or a surveyor could calculate the area of a large tract of land without having to measure each side of the tract.

In the context of the exercise, we use the angle of depression and the height of the lighthouse to determine the distance of a boat from the shore. Mariners, pilots, and even wildlife biologists often use these concepts to find distances to objects when direct measurement is impossible. Thus, understanding how to apply trigonometric ratios, like the tangent, to real-world problems is a valuable skill across many professions.