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Finding Heights and Distances

🔄 Quick Recap​

In the previous section, we learned about angles of elevation and depression. Now, we'll use these angles along with trigonometric ratios to find heights and distances.

📚 The Basic Approach​

When we want to find the height of a tall object (like a building, tower, or tree) or the distance between two points, we form a right-angled triangle and use trigonometric ratios.

Here's our strategy:

  1. Identify what we know and what we need to find
  2. Draw a diagram with a right-angled triangle
  3. Choose the appropriate trigonometric ratio
  4. Substitute the values and solve

Height and Distance Formula

🧮 Mathematical Corner​

Let's look at the basic formulas we use:

For finding height when we know the distance and angle of elevation:

tan θ = height/distance
height = distance × tan θ

For finding distance when we know the height and angle of elevation:

tan θ = height/distance
distance = height ÷ tan θ

For angle of depression, the formula is the same because the angle of depression is equal to the angle of elevation at the same points.

🌍 Real-Life Applications​

These calculations are used in many real-world situations:

  1. Architecture: Engineers calculate heights and distances when designing buildings and bridges.
  2. Navigation: Sailors and pilots use angles and distances to find their position.
  3. Astronomy: Scientists measure angles to calculate distances to stars and planets.
  4. Photography: Photographers use angles to frame their shots perfectly.
  5. Construction: Workers use these principles when building structures.

✅ Solved Example 1: Height of a Tower​

Problem: A tower stands vertically on the ground. From a point on the ground, which is 15 m away from the foot of the tower, the angle of elevation of the top of the tower is found to be 60°. Find the height of the tower.

Solution:

Step 1: Let's understand what we know

  • Distance from the observer to the tower = 15 m
  • Angle of elevation = 60°
  • We need to find the height of the tower

Step 2: Draw a right-angled triangle

  • Let's call the height of the tower as h
  • The base of the triangle is 15 m
  • The angle at the base is 60°

Step 3: Choose the appropriate trigonometric ratio

  • Since we know the angle and the adjacent side (15 m) and need to find the opposite side (height), we use the tangent ratio.
  • tan θ = opposite/adjacent

Step 4: Substitute and solve

tan 60° = h/15
h = 15 × tan 60°
h = 15 × √3 (since tan 60° = √3)
h = 15 × 1.732
h = 25.98 m

Therefore, the height of the tower is approximately 26 m.

✅ Solved Example 2: Distance to a Building​

Problem: The angle of elevation of the top of a building from a point on the ground is 30°. If the observer is 20 m from the building, find the height of the building.

Solution:

Step 1: Let's understand what we know

  • Distance from the observer to the building = 20 m
  • Angle of elevation = 30°
  • We need to find the height of the building

Step 2: Draw a right-angled triangle

  • Let's call the height of the building as h
  • The base of the triangle is 20 m
  • The angle at the base is 30°

Step 3: Choose the appropriate trigonometric ratio

  • We use tangent again
  • tan θ = opposite/adjacent

Step 4: Substitute and solve

tan 30° = h/20
h = 20 × tan 30°
h = 20 × (1/√3) (since tan 30° = 1/√3)
h = 20 × 0.5774
h = 11.55 m

Therefore, the height of the building is approximately 11.55 m.

🧪 Activity Time!​

Make Your Own Height Calculator!

Materials needed:

  • A straw or pencil
  • String
  • A small weight (like a paper clip)
  • A protractor

Steps:

  1. Tie the string with the weight to the middle of the straw
  2. Hold the straw and look through it at the top of a tall object (like a tree)
  3. Ask a friend to measure the angle made by the string with the vertical
  4. Measure your distance from the object
  5. Use the formula: height = distance × tan(angle) to calculate the height!

⚠️ Common Misconceptions​

  1. Misconception: We always use the tangent ratio in height and distance problems. Correction: The ratio we use depends on what we know and what we need to find. Sometimes sine or cosine is more appropriate.

  2. Misconception: The observer's height doesn't matter. Correction: In precise calculations, we often need to add the observer's height to get the total height.

🧠 Memory Tricks​

Remember which ratio to use with the word "TOA" from SOH-CAH-TOA:

  • T: Tangent
  • O: Opposite
  • A: Adjacent

So if you need to find the opposite side (height) and know the adjacent side (distance), use tangent!

💡 Key Points to Remember​

  • Always draw a clear diagram of the right-angled triangle.
  • Label what you know and what you need to find.
  • Choose the appropriate trigonometric ratio based on what you know.
  • Remember to consider the observer's height in the final answer if needed.
  • Double-check your calculations, especially when using calculators.

🤔 Think About It!​

If you're at the top of a lighthouse and see a ship with an angle of depression of 30°, how does the distance to the ship change if the angle of depression changes to 45°? Is the ship moving closer or farther away?

🔜 What Next?​

Now that we understand how to solve basic height and distance problems, we'll look at more complex scenarios involving multiple angles or combined measurements.