Practice Problems

🔄 Quick Recap

We've learned how to use trigonometry to find heights and distances in various scenarios. Now, it's time to practice these skills with more problems.

📚 How to Approach These Problems

Remember our problem-solving strategy:

1. Read the problem carefully and identify what is given and what you need to find
2. Draw a clear diagram showing the situation
3. Label all known values on your diagram
4. Choose the appropriate trigonometric ratio(s)
5. Set up your equation(s) and solve
6. Check if your answer makes sense

✅ Solved Examples with Explanations

Let's work through a few examples in detail to help you understand the approach better.

Example 1: The Circus Rope

Problem: A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole if the angle made by the rope with the ground level is 30°.

Solution:

Step 1: Understand what we know

• Length of the rope = 20 m
• Angle between the rope and the ground = 30°
• We need to find the height of the pole

Step 2: Draw and analyze the diagram

• The rope forms the hypotenuse of a right-angled triangle
• The pole forms the opposite side to the 30° angle
• The ground forms the adjacent side

Step 3: Choose the appropriate ratio

• Since we know the hypotenuse (rope length) and need to find the opposite side (pole height), we use the sine ratio.

Step 4: Solve

sin 30° = height of pole/length of rope
sin 30° = height/20
height = 20 × sin 30°
height = 20 × 0.5
height = 10 m

Therefore, the height of the pole is 10 meters.

Example 2: The Slide for Children

Problem: A contractor plans to install two slides for children to play in a park. For children below the age of 5 years, she prefers to have a slide whose top is at a height of 1.5 m and is inclined at an angle of 30° to the ground. For elder children, she wants to have a steeper slide at a height of 3 m, inclined at an angle of 60° to the ground. What should be the length of the slide in each case?

Solution:

For the slide for younger children:

• Height of the slide = 1.5 m
• Angle of inclination = 30°
• We need to find the length of the slide (the hypotenuse)

Using the sine ratio:

sin 30° = 1.5/length
0.5 = 1.5/length
length = 1.5/0.5
length = 3 m

For the slide for older children:

• Height of the slide = 3 m
• Angle of inclination = 60°
• We need to find the length of the slide (the hypotenuse)

Using the sine ratio:

sin 60° = 3/length
0.866 = 3/length
length = 3/0.866
length = 3.46 m

Therefore, the slide for younger children should be 3 m long, and the slide for older children should be 3.46 m long.

🧪 Activity Time!

Make Your Own Practice Problems

Try creating your own height and distance problems based on objects around you:

1. Measure how far you stand from a tall object (like a tree or flagpole)
2. Estimate the angle of elevation to its top
3. Calculate its height using the formula: height = distance × tan(angle)
4. If possible, verify your answer by measuring the actual height

🧮 Practice Problems

Now try these problems on your own. Draw diagrams and solve step by step.

Problem 1

A kite is flying at a height of 60 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is 60°. Find the length of the string, assuming there is no slack.

Problem 2

From a point on the ground, the angle of elevation of the top of a tower is 30°. If the observer is 30 m away from the foot of the tower, find the height of the tower.

Problem 3

A tree breaks due to a storm, and the broken part bends so that the top of the tree touches the ground, making an angle of 30° with it. The distance between the foot of the tree and the point where the top touches the ground is 8 m. Find the height of the tree.

Problem 4

A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from 30° to 60° as he walks towards the building. Find the distance he walked towards the building.

Problem 5

A statue, 1.6 m tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60° and from the same point, the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal.

💡 Hints for Solving

Hint for Problem 1

The string forms the hypotenuse of a right-angled triangle. Use the sine of 60° to find the relationship between the height and the string length.

Hint for Problem 2

Use the tangent of 30° to relate the height of the tower to the distance from the observer.

Hint for Problem 3

This problem requires multiple steps. First, find the height of the standing part of the tree using the tangent of 30°. Then, use the Pythagorean theorem to find the length of the broken part.

Hint for Problem 4

Use the tangent of both angles (30° and 60°) to create two equations with the distance variables. The difference in these distances is what he walked.

Hint for Problem 5

Let the height of the pedestal be h. Then the total height of the statue and pedestal is (h + 1.6). Use the two angles of elevation to create two equations and solve for h.

🧠 Memory Tricks for Important Values

Remember these values to help you solve problems more quickly:

• sin 30° = 1/2 = 0.5
• sin 45° = 1/√2 = 0.7071
• sin 60° = √3/2 = 0.866
• cos 30° = √3/2 = 0.866
• cos 45° = 1/√2 = 0.7071
• cos 60° = 1/2 = 0.5
• tan 30° = 1/√3 = 0.5774
• tan 45° = 1
• tan 60° = √3 = 1.732

⚠️ Common Mistakes to Avoid

1. Not identifying the correct triangle: Make sure you correctly identify the right-angled triangle in the problem.

2. Using the wrong trigonometric ratio: Match the ratio to what you know and what you need to find.

3. Not considering the observer's height: In some problems, you need to add the observer's height to get the total height.

4. Mixing up angles of elevation and depression: Remember that for the same line of sight, the angle of elevation and angle of depression are equal.

5. Calculation errors: Double-check your arithmetic, especially when using a calculator.

🔜 What Next?

After practicing these problems, you'll be ready to tackle the end-of-chapter exercises with confidence. Remember, the key to mastering trigonometry is practice and careful application of the concepts you've learned.