Word Problems with Angle of Elevation and Depression - Complete Guide

Master angle of elevation and depression through these 3 detailed exercises with comprehensive solutions, definitions, rules, and visual aids.

Exercises and Solutions
1 Angle of Elevation
Exercise 1
Sarah is standing 30 meters away from a tall tree. She looks up at the top of the tree and measures an angle of elevation of 45°. How tall is the tree?
Definition:

Angle of Elevation: The angle formed between the horizontal line and the line of sight when looking upward.

Method:
  1. Identify the horizontal distance (adjacent side)
  2. Identify the angle of elevation
  3. Use trigonometry to find the height (opposite side)
  4. Apply the tangent ratio: tan(angle) = opposite/adjacent
30m Tree Height 45°
Step 1: Identify given information

Distance from tree = 30 meters

Angle of elevation = 45°

We need to find: Height of the tree

Step 2: Set up the trigonometric relationship

tan(45°) = Height ÷ Distance

tan(45°) = Height ÷ 30

Step 3: Solve for the height

Since tan(45°) = 1

1 = Height ÷ 30

Height = 30 × 1 = 30 meters

The tree is 30 meters tall
Final answer:

The tree is 30 meters tall.

Applied rules:

• Angle of Elevation: Always measured from the horizontal upward

• Tangent Ratio: tan(θ) = opposite/adjacent

• Special Angle: tan(45°) = 1

2 Angle of Depression
Exercise 2
From the top of a 20-meter building, a person looks down at a car on the street below. The angle of depression to the car is 30°. How far is the car from the base of the building?
Definition:

Angle of Depression: The angle formed between the horizontal line and the line of sight when looking downward.

Method:
  1. Identify the vertical height (opposite side)
  2. Identify the angle of depression
  3. Use the fact that angle of depression equals angle of elevation from the car's perspective
  4. Apply the tangent ratio: tan(angle) = opposite/adjacent
20m Distance = ? 30°
Step 1: Identify given information

Height of building = 20 meters

Angle of depression = 30°

We need to find: Distance from building to car

Step 2: Understand the relationship

Angle of depression from top = Angle of elevation from car = 30°

This creates a right triangle where:

- Opposite side = height of building = 20m

- Adjacent side = distance to car = ?

- Angle = 30°

Step 3: Set up the trigonometric relationship

tan(30°) = Height ÷ Distance

tan(30°) = 20 ÷ Distance

Step 4: Solve for the distance

Since tan(30°) = √3/3 ≈ 0.577

√3/3 = 20 ÷ Distance

Distance = 20 ÷ (√3/3) = 20 × (3/√3) = 60/√3

Rationalizing: Distance = (60/√3) × (√3/√3) = 60√3/3 = 20√3 ≈ 34.64 meters

The car is approximately 34.64 meters from the base of the building
Final answer:

The car is approximately 34.64 meters from the base of the building.

Applied rules:

• Angle of Depression: Equal to angle of elevation from the object's perspective

• Tangent Ratio: tan(θ) = opposite/adjacent

• Special Angle: tan(30°) = √3/3

3 Combined Problem
Exercise 3
A bird is flying at a height of 50 meters above the ground. It spots a fish in a pond and sees it at an angle of depression of 60°. How far is the fish from the point directly below the bird?
Definition:

Horizontal Distance: The distance along the ground from one point to another, ignoring vertical differences.

Method:
  1. Identify the vertical height (altitude)
  2. Identify the angle of depression
  3. Set up a right triangle with the height as one leg
  4. Use trigonometry to find the horizontal distance
50m Distance = ? 60°
Step 1: Identify given information

Height of bird = 50 meters

Angle of depression = 60°

We need to find: Horizontal distance to fish

Step 2: Analyze the triangle formed

This creates a right triangle where:

- Vertical side (height) = 50 meters

- Horizontal side (distance) = unknown

- Angle of depression = 60°

Step 3: Set up the trigonometric relationship

tan(60°) = Height ÷ Distance

tan(60°) = 50 ÷ Distance

Step 4: Solve for the distance

Since tan(60°) = √3 ≈ 1.732

√3 = 50 ÷ Distance

Distance = 50 ÷ √3

Rationalizing: Distance = (50/√3) × (√3/√3) = 50√3/3 ≈ 28.87 meters

The fish is approximately 28.87 meters from the point directly below the bird
Final answer:

The fish is approximately 28.87 meters from the point directly below the bird.

Applied rules:

• Angle of Depression: Creates a right triangle with horizontal and vertical distances

• Tangent Ratio: Essential for solving angle-distance relationships

• Special Angle: tan(60°) = √3

Key Concepts, Rules, and Methods
tan(θ) = Opposite/Adjacent
Tangent Ratio in Right Triangles
Key Definitions:

Angle of Elevation: The angle between the horizontal and the line of sight when looking up.

Angle of Depression: The angle between the horizontal and the line of sight when looking down.

Line of Sight: The straight line from the observer's eye to the observed object.

Horizontal Line: A line parallel to the ground level.

Step-by-Step Method:
  1. Draw a diagram: Sketch the situation showing all known measurements
  2. Identify the triangle: Locate the right triangle formed by the problem
  3. Label sides: Identify the opposite, adjacent, and hypotenuse
  4. Choose ratio: Select the appropriate trigonometric ratio
  5. Set up equation: Write the trigonometric equation
  6. Solve: Calculate the unknown measurement
Important Rule: The angle of depression from the observer equals the angle of elevation from the object being observed.
Memory Aid: "Elevation = Looking UP, Depression = Looking DOWN"
Key Formula: In right triangles, tan(angle) = opposite/adjacent
Tip 1: Always draw a diagram first to visualize the problem.
Tip 2: Remember that the horizontal line is always level with the ground.
Tip 3: Check if your answer makes sense in the context of the problem.
Special Angles:
tan(30°) = √3/3 ≈ 0.577
tan(45°) = 1
tan(60°) = √3 ≈ 1.732
Common Mistakes to Avoid:

• Confusing angle of elevation with angle of depression

• Misidentifying the opposite and adjacent sides

• Forgetting to use the horizontal line as the reference

• Not drawing accurate diagrams

Questions & Answers

Question: Why do we use the tangent ratio for these problems instead of sine or cosine?

Answer: Great question! We typically use the tangent ratio because these angle of elevation/depression problems often involve:

  • The angle (which we measure)
  • The horizontal distance (what we walk/run)
  • The vertical distance (the height)

In our right triangles, we usually know the angle and one of the legs (horizontal or vertical), and we want to find the other leg. The tangent ratio relates the opposite and adjacent sides directly: tan(θ) = opposite/adjacent.

For example, if we know the angle of elevation and the horizontal distance, we can find the height using: Height = Distance × tan(angle).

Question: How can I remember which is angle of elevation and which is angle of depression?

Answer: Here's a simple memory trick:

  • Elevation = Going UP: Think "ELEPHANT going UP the mountain" - you look up to see an elephant on a hill
  • Depression = Going DOWN: Think "DEEP in the valley" - you look down into a deep valley

Another way to remember: "E" comes before "D" in the alphabet, just like UP comes before DOWN. So Elevation = Up, Depression = Down.

Always remember that both angles are measured from the HORIZONTAL line, not from the vertical!

Question: What if I get a negative answer when solving these problems? Does that make sense?

Answer: No, negative answers don't make sense for distances in these problems! Here's why:

  • Distances are always positive measurements
  • Heights and lengths cannot be negative in real-world contexts
  • If you get a negative answer, you likely made a calculation error

Common causes of negative results:

  • Using the wrong trigonometric ratio
  • Mistaking the angle (using depression when it's elevation, etc.)
  • Division errors or calculator mistakes

Always check: Does your answer seem reasonable? If a tree is 30 meters tall and you calculated it as 300 meters, reconsider your work!