Hey guys! Let's dive into the solutions for Chapter 23 of your ICSE Class 10 Maths textbook. This chapter is super important, and mastering it will definitely boost your confidence for the exams. We'll break down each concept, making sure you understand everything thoroughly. So, grab your notebooks, and let's get started!

Understanding the Basics

Before we jump into the solutions, let's quickly recap the fundamental concepts covered in Chapter 23. This chapter typically revolves around topics like trigonometry, heights and distances, and related applications. Trigonometry is all about the relationships between the angles and sides of triangles, especially right-angled triangles. You'll need to be familiar with trigonometric ratios such as sine, cosine, tangent, cosecant, secant, and cotangent. These ratios are the building blocks for solving problems related to angles of elevation and depression, which are key in height and distance problems.

Trigonometric Ratios: Make sure you know the definitions of sin θ, cos θ, and tan θ. Remember, sin θ = Opposite/Hypotenuse, cos θ = Adjacent/Hypotenuse, and tan θ = Opposite/Adjacent. Also, understand that cosec θ, sec θ, and cot θ are the reciprocals of sin θ, cos θ, and tan θ, respectively.

Angles of Elevation and Depression: The angle of elevation is the angle formed between the horizontal line and the line of sight when you look upwards at an object. The angle of depression is the angle formed when you look downwards. Visualizing these angles correctly is crucial for setting up your equations.

Applications in Heights and Distances: These problems usually involve finding the height of objects like buildings or trees, or the distance between objects, using trigonometric ratios and angles of elevation or depression. The key here is to draw accurate diagrams and identify the right triangles to apply the trigonometric ratios.

Why is this chapter so important? Well, trigonometry and its applications are not just confined to your textbook. They have real-world applications in fields like engineering, navigation, and surveying. Understanding these concepts will not only help you score well in your exams but also build a strong foundation for future studies in science and mathematics.

To really nail this chapter, practice is key. Work through as many problems as you can, and don't hesitate to ask your teachers or friends for help if you get stuck. Remember, every problem you solve is a step closer to mastering the concepts.

Detailed Solutions and Explanations

Now, let's dive into some example problems and their solutions. We'll go through each step in detail so you can understand the logic behind the solutions. Let's tackle some common types of questions you might encounter.

Example 1: Finding the Height of a Tower

Problem: From the top of a cliff 20 meters high, the angle of depression of a boat is 60°. Find the distance of the boat from the foot of the cliff.

Solution:

1. Draw a Diagram: Always start by drawing a diagram. This will help you visualize the problem and identify the relevant triangles.
2. Identify the Right Triangle: In this case, we have a right triangle formed by the cliff, the horizontal line from the top of the cliff to the boat's position, and the line of sight from the top of the cliff to the boat.
3. Label the Diagram: Label the height of the cliff as 20 meters. The angle of depression is 60°, which means the angle of elevation from the boat to the top of the cliff is also 60° (alternate angles).
4. Choose the Correct Trigonometric Ratio: We need to find the distance of the boat from the foot of the cliff, which is the adjacent side to the angle of elevation. We know the opposite side (height of the cliff). Therefore, we'll use the tangent ratio: tan θ = Opposite/Adjacent.
5. Set up the Equation: tan 60° = 20 / Distance. We know that tan 60° = √3. So, √3 = 20 / Distance.
6. Solve for the Distance: Distance = 20 / √3. To rationalize the denominator, multiply both the numerator and denominator by √3. Distance = (20√3) / 3 meters.

Therefore, the distance of the boat from the foot of the cliff is (20√3) / 3 meters. Make sure to include the units in your final answer.

Example 2: Finding the Angle of Elevation

Problem: A ladder 15 meters long just reaches the top of a vertical wall. If the ladder makes an angle of 60° with the wall, find the height of the wall.

Solution:

1. Draw a Diagram: Draw a vertical wall and a ladder leaning against it. The ladder forms the hypotenuse of a right triangle.
2. Identify the Right Triangle: The right triangle is formed by the wall, the ground, and the ladder.
3. Label the Diagram: Label the length of the ladder as 15 meters. The angle between the ladder and the wall is 60°.
4. Choose the Correct Trigonometric Ratio: We need to find the height of the wall, which is the adjacent side to the 60° angle. We know the hypotenuse (length of the ladder). Therefore, we'll use the cosine ratio: cos θ = Adjacent/Hypotenuse.
5. Set up the Equation: cos 60° = Height / 15. We know that cos 60° = 1/2. So, 1/2 = Height / 15.
6. Solve for the Height: Height = 15 / 2 = 7.5 meters.

Therefore, the height of the wall is 7.5 meters. Always double-check your calculations and units.

Example 3: Problems Involving Two Angles of Elevation

Problem: From the top of a building 20 m high, the angle of elevation of the top of a tower is 60° and the angle of depression of the foot of the tower is 30°. What is the height of the tower?

Solution:

1. Draw a Diagram: Draw the building and the tower. The angles of elevation and depression are formed from the top of the building to the top and foot of the tower, respectively.
2. Identify the Right Triangles: There are two right triangles in this problem. One is formed by the horizontal line from the top of the building to the tower, the height difference between the building and tower, and the line of sight to the top of the tower. The other is formed by the horizontal line, the height of the building, and the line of sight to the foot of the tower.
3. Label the Diagram: Label the height of the building as 20 m. The angle of elevation is 60°, and the angle of depression is 30°.
4. Find the Distance Between the Building and Tower: Using the angle of depression, we can find the horizontal distance between the building and the tower. tan 30° = 20 / Distance. Since tan 30° = 1/√3, we have 1/√3 = 20 / Distance. So, Distance = 20√3 meters.
5. Find the Height Difference: Now, use the angle of elevation to find the height difference between the top of the building and the top of the tower. tan 60° = Height Difference / Distance. Since tan 60° = √3, we have √3 = Height Difference / (20√3). So, Height Difference = √3 * 20√3 = 60 meters.
6. Find the Total Height of the Tower: The total height of the tower is the height of the building plus the height difference. Total Height = 20 + 60 = 80 meters.

Therefore, the height of the tower is 80 meters. These types of problems require careful diagram drawing and application of trigonometric ratios.

Tips and Tricks for Solving Problems

To excel in solving problems from this chapter, here are some tips and tricks that can help you:

• Draw Accurate Diagrams: A well-drawn diagram is half the battle won. Make sure to label all the given information correctly.
• Identify the Right Triangles: Look for right triangles in the problem. These are the key to applying trigonometric ratios.
• Choose the Correct Trigonometric Ratio: Decide which trigonometric ratio (sin, cos, tan) is appropriate based on the given information and what you need to find.
• Memorize Trigonometric Values: Knowing the values of trigonometric ratios for standard angles (0°, 30°, 45°, 60°, 90°) will save you time.
• Practice Regularly: The more you practice, the better you'll become at recognizing patterns and applying the correct formulas.
• Check Your Answers: Always double-check your calculations and make sure your answer makes sense in the context of the problem.

Common Mistakes to Avoid

Even with a good understanding of the concepts, it's easy to make mistakes. Here are some common mistakes to watch out for:

• Incorrectly Labeling the Diagram: Make sure you label the angles and sides correctly. A wrong label can lead to an incorrect solution.
• Using the Wrong Trigonometric Ratio: Choosing the wrong ratio can completely change your answer. Double-check which sides and angles you're working with.
• Forgetting Units: Always include the units in your final answer. For example, if you're finding a distance, make sure to include meters or centimeters.
• Not Rationalizing the Denominator: If your answer has a square root in the denominator, rationalize it to simplify the expression.
• Rounding Off Too Early: Avoid rounding off intermediate calculations. Round off only at the end to get the most accurate answer.

Conclusion

So, there you have it! A comprehensive guide to tackling Chapter 23 of your ICSE Class 10 Maths textbook. Remember, practice is the key to mastering these concepts. Work through as many problems as you can, and don't be afraid to ask for help when you need it. With a solid understanding of the fundamentals and plenty of practice, you'll be well on your way to acing your exams. Good luck, guys! You got this!