Lesson 4: Angles of Elevation and Depression

👨🏽‍🏫 Introduction

Have you ever looked up at a tall building or down from a hill and wondered how far away the base or top is? 🏙️⛰️ These situations involve angles of elevation and depression—powerful tools in trigonometry for solving real-world height and distance problems!

By the end of this lesson, you’ll know how to model these situations using right-angled triangles and solve them using sine, cosine, or tangent. 📐🧮

• 👁 Understand angles of elevation and depression
• 📏 Use right triangles to model real-world problems
• 📐 Apply trig ratios to calculate heights and distances

🧠 Key Concepts

📌 1. Angle of Elevation

The angle of elevation is the angle formed when you look upward from a horizontal line to an object above.

📌 2. Angle of Depression

The angle of depression is the angle formed when you look downward from a horizontal line to an object below.

In both cases, the triangle formed is a right-angled triangle, and the horizontal line is adjacent to the angle.

📌 3. Choosing the Right Ratio

• Use tan(θ) = opposite / adjacent when you know or need height and horizontal distance
• Use sin(θ) or cos(θ) if the hypotenuse is involved

🔍 Worked Examples

1. 1. Example 1 – Angle of Elevation:
      A tree is 20 m tall. You stand 15 m away and look up at the top. Find the angle of elevation θ.
      
      
      🟢 Answer: 53°

1. 1. Example 2 – Angle of Depression:
      A drone is flying 50 m above the ground and spots a person 120 m away horizontally. Find the angle of depression.
      
      
      🟢 Answer: 22.6°

1. Example 3 – Find height using angle of elevation:
   From a point 30 m away from a tower, the angle of elevation to the top is 60°. Find the height.
   
   🟢 Answer: Approximately 51.96 m

✍🏽 Practice Exercises

🅰️ Section A – Concept Review

1. Define the angle of elevation and angle of depression.
2. Which trig ratio is commonly used for solving elevation and depression problems?

🅱️ Section B – Application Problems

1. You look up at a window 12 m above the ground. If you’re standing 9 m from the building, find the angle of elevation.
2. A lighthouse is 25 m tall. A boat is 70 m from the base. Find the angle of elevation to the top of the lighthouse.

🅾️ Section C – Mixed Practice

1. A pilot descends toward a runway from a height of 300 m at an angle of depression of 10°. How far is the plane horizontally from the runway?

✅ Answers and Explanations

• A1:
  – Elevation: angle above horizontal line
  – Depression: angle below horizontal line
• A2: Tangent (tan) is most commonly used.
• B1:
• B2:
• C1:

📌 Recap

• ⬆️ Angle of elevation = looking up
• ⬇️ Angle of depression = looking down
• 📐 Use trig ratios (usually tangent) with right triangles to solve for height or distance

💭 Reflection Prompt

Look at a tall structure like a flagpole, tower, or billboard. Estimate your distance from it and imagine the angle you’d see it at. Could you calculate its height?