CBSE 10 Trigonometry: A Must-Have Guide to Conquer Heights and Distances (Examples Included)

We know that heights and distances are a fundamental concept in trigonometry. In this article, we will explore what heights and distances are, how to find them, and the formulas and techniques that you need to know. We will also go through an example question and practice problems to solidify your understanding.

What are heights and distances?

Heights and distances are used to find the height or distance of an object that is difficult or impossible to measure directly. For example, if you are standing at the base of a tree, you may want to find the height of the tree without climbing it. Or, if you are standing on the shore of a lake, you may want to find the distance to an object on the other side of the lake.

How to dind heights and distances?

To find heights and distances, we use trigonometric ratios: sine, cosine, and tangent. These ratios relate the angles and sides of a right-angled triangle. By measuring one or more angles and one side of the triangle, we can find the other sides.

Heights and distances formulas

Here are some important formulas that you need to know for heights and distances:

• The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse: sin θ = opposite / hypotenuse
• The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse: cos θ = adjacent / hypotenuse
• The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side: tan θ = opposite / adjacent
• The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides: (hypotenuse)² = (opposite)² + (adjacent)²

Heights and distances question

Let's work through an example to see how we can use these formulas to find heights and distances.

Example:

You are standing 20 meters away from a building. You measure the angle of elevation to the top of the building to be 30°. What is the height of the building?

✅  Solution:
      1. We can use the tangent ratio to solve this problem.
      2. Let h be the height of the building.
      3. Then tan 30° = h / 20.
      4. Solving for h, we get h = 20 * tan 30° = 20 / √3 meters.

Heights and distances practice problems

Now that we have gone through an example, it's time to practice on your own. Try solving these practice problems:

• You are standing 50 meters away from a flagpole. The angle of elevation to the top of the flagpole is 45°. What is the height of the flagpole?
• You are standing on the bank of a river. You measure the angle of elevation to a tree on the other side of the river to be 60°. The distance between you and the tree is 100 meters. How wide is the river?

FAQs

What are some real-world applications of heights and distances?

Heights and distances are used in a wide range of fields, including architecture, engineering, and surveying. They can be used to determine the height of buildings, the distance between two points, and the angle of elevation or depression of an object.

Can heights and distances be calculated without using trigonometric ratios?

While trigonometric ratios are the most common method for calculating heights and distances, there are other methods that can be used depending on the specific problem. For example, if the angle of elevation is 45°, the height and distance will be equal.

How do I know whether to use sine, cosine, or tangent to solve a problem?

The trigonometric ratio you use will depend on the information you have been given in the problem. If you know the length of the opposite and hypotenuse, you can use sine. If you know the length of the adjacent and hypotenuse, you can use cosine. If you know the length of the opposite and adjacent, you can use tangent.

What is the Pythagorean Theorem, and how is it used in heights and distances problems?

The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This theorem can be used to find the length of one side of a triangle if the lengths of the other two sides are known.

Conclusion

Heights and distances are an important concept in trigonometry that can be used to find the height or distance of an object that is difficult or impossible to measure directly. By using trigonometric ratios and the Pythagorean Theorem, we can solve a wide range of problems related to heights and distances. If you want to practice more, check out Aha AI for additional resources and practice problems.

Relevant links

• CBSE 10 - Trigonometry - Trigonometric Identities
• Heights and Distances by BYJU'S
• Heights and Distances by Vedantu
• NCERT Solutions for Class 10 Mathematics - Heights and Distances
• Heights and Distances by CUEMATH