Angles of Elevation and Depression

Jul 15, 2024

Lecture Notes: Angles of Elevation and Depression

Introduction

  • Presenter: Mr. A
  • Topic: Angles of Elevation vs. Angles of Depression
  • Relevance: Common topic in trigonometry and geometry

Definitions

  • Angle of Elevation: Looking up from horizontal
  • Angle of Depression: Looking down from horizontal

Key Concepts

  • Relative Terms: Both angles are relative; no absolute 'up' or 'down'
  • Example Illustration:
    • Lighthouse and Friend: Lighthouse as observer (horizontal line at the top), friend at the bottom.
    • Angle of Depression: Angle between the horizontal line from the observer to the friend below.
    • Angle of Elevation: Friend looking up at the observer at the same angle (alternate interior angles).

Mathematical Application

Example 1: Bald Eagle

  • Scenario: Spotting a bald eagle in a tree.
  • Given: Angle of elevation = 67 degrees, distance from tree = 150 feet.
  • To Find: Height of the eagle.
  • Steps:
    1. Draw a right triangle.
    2. Label given values and unknown height (x).
    3. Use trigonometric function:
      • \( \tan(67^\circ) = \frac{x}{150} \
      • \( x = 150 \times \tan(67^\circ) \
    4. Calculation: x ≈ 353 feet.

Example 2: Angle of Depression - Friend from Building

  • Scenario: Looking down from a building to a friend.
  • Given: Angle of depression = 57 degrees, height of building = 200 feet.
  • To Find: Distance of friend from the building.
  • Steps:
    1. Draw the horizontal line and right triangle.
    2. Consider alternate interior angles for depression angle inside the triangle.
    3. Use trigonometry:
      • \( \tan(57^\circ) = \frac{200}{x} \
      • \( x = \frac{200}{\tan(57^\circ)} \
    4. Calculation: x ≈ 130 feet.

Example 3: Flying a Kite

  • Scenario: Friend observes angle of elevation (63 degrees) to a kite.
  • Given: Kite height = 100 feet, hand height = 5 feet.
  • To Find: Length of string.
  • Steps:
    1. Adjust height: effective triangle height = 95 feet (100 - 5).
    2. Use trigonometry:
      • \( \sin(63^\circ) = \frac{95}{x} \
      • \( x = \frac{95}{\sin(63^\circ)} \
    3. Calculation: x ≈ 107 feet.

Finding Missing Angles

Example 4: Elevation Angle to Building Flag

  • Scenario: Looking up at a flag on a 50-foot building from 800 feet away.
  • Given: Building height = 50 feet, distance = 800 feet.
  • To Find: Angle of elevation.
  • Steps:
    1. Draw right triangle with given values.
    2. Use inverse tangent:
      • \( \tan(\theta) = \frac{50}{800} \
      • \( \theta = \arctan(\frac{50}{800}) \
    3. Calculation: θ ≈ 3.6 degrees.

Example 5: Depression Angle from Hot Air Balloon

  • Scenario: Observing house from a hot air balloon.
  • Given: Balloon height = 1,000 feet, distance to house = 10,000 feet.
  • To Find: Angle of depression.
  • Steps:
    1. Draw horizontal line and right triangle.
    2. Consider alternate interior angles.
    3. Use inverse sine:
      • \( \sin(\theta) = \frac{1000}{10000} \
      • \( \theta = \arcsin(\frac{1000}{10000}) \
    4. Calculation: θ ≈ 5.74 degrees.

Key Takeaways

  • Always consider horizontal lines when dealing with angles of elevation and depression.
  • Practice using trigonometric functions and their inverses.

Closing

  • Reminder: Draw accurate diagrams to avoid mistakes.
  • Call to Action: Like, subscribe, comment, and have a great day!