🔍 Introduction

Trigonometry is not just theoretical—it’s highly practical! This chapter shows how to apply trigonometric concepts to real-life problems like measuring heights and distances where direct measurement isn’t possible.

🏔️ 1. Line of Sight, Angle of Elevation & Depression

🔹 Line of Sight

The imaginary line between the observer's eye and the object being viewed.

🔹 Angle of Elevation

The angle formed when the observer looks up at an object above the horizontal level.

🔹 Angle of Depression

The angle formed when the observer looks down at an object below the horizontal level.

📌 Angles are always measured with respect to the horizontal line.

📐 2. Key Terminologies

  • Observer – The person viewing the object.

  • Object – The item whose height or distance is to be measured.

  • Base – Horizontal distance between observer and object.

  • Height – Vertical distance to be found using trigonometry.

📊 3. Application of Trigonometric Ratios

Using trigonometric ratios:

sin⁡θ=PerpendicularHypotenuse,cos⁡θ=BaseHypotenuse,tan⁡θ=PerpendicularBase\sin \theta = \frac{\text{Perpendicular}}{\text{Hypotenuse}}, \quad \cos \theta = \frac{\text{Base}}{\text{Hypotenuse}}, \quad \tan \theta = \frac{\text{Perpendicular}}{\text{Base}}

These help solve problems involving:

  • Heights of towers

  • Distance across rivers

  • Heights of buildings seen from a distance

📌 4. Common Scenarios

🧱 Example 1: Height of a building

A boy observes the top of a building at an angle of elevation of 30°. If he is standing 50 m from the building, find its height.

Solution:

tan⁡30∘=h50,13=h50⇒h=503≈28.87 m\tan 30^\circ = \frac{h}{50}, \quad \frac{1}{\sqrt{3}} = \frac{h}{50} \Rightarrow h = \frac{50}{\sqrt{3}} \approx 28.87 \, \text{m}

🛳️ Example 2: Distance from the shore

A lighthouse is 45 m high. The angle of depression to a boat is 60°. Find the distance of the boat from the lighthouse base.

Solution:

tan⁡60∘=45x,3=45x⇒x=453≈25.98 m\tan 60^\circ = \frac{45}{x}, \quad \sqrt{3} = \frac{45}{x} \Rightarrow x = \frac{45}{\sqrt{3}} \approx 25.98 \, \text{m}

🧠 Tips for Solving Problems

  • Always draw a neat diagram.

  • Label the triangle: height, base, and angle.

  • Use correct trigonometric ratio based on given values.

  • Convert final answers to two decimal places.

📝 Things to Remember

  • tan θ = perpendicular / base is used most frequently.

  • Trigonometry is applied only in right-angled triangles.

  • Angles should be measured from the horizontal.

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