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

🧮 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.