How to Solve Height and Distance Problems Using Trigonometry
The concept of Application of Trigonometry plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Whether you need to find the height of a building, measure the width of a river, or solve board exam problems, applications of trigonometry make all these tasks easier. Understanding this topic not only sharpens your calculation skills but also shows you how maths can be useful in daily life.
What Is Application of Trigonometry?
The Application of Trigonometry is all about using the properties and ratios of right-angled triangles (sine, cosine, tangent, etc.) to solve practical problems involving heights, distances, and angles. You’ll find this concept applied in areas such as architecture, navigation, and engineering.
Key Formula for Application of Trigonometry
Here’s the standard formula: \( \text{For a right triangle,} \ \sin \theta = \frac{\text{Opposite side}}{\text{Hypotenuse}}, \ \cos \theta = \frac{\text{Adjacent side}}{\text{Hypotenuse}}, \ \tan \theta = \frac{\text{Opposite side}}{\text{Adjacent side}} \)
Cross-Disciplinary Usage
Application of trigonometry is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for exams like NTSE, JEE, or NEET will see its relevance in solving distance, angle, and measurement questions.
Where Do We Use Application of Trigonometry?
Applications of trigonometry are seen in many real-life and academic scenarios:
- Measuring the height of tall structures without direct access
- Calculating distance across rivers or valleys
- Navigation and mapping (used in GPS and ships)
- Designing engineering structures (bridges, ramps)
- Astronomy and physics calculations
Typical Application Problems in Class 10
In class 10, you mainly handle height and distance problems using trigonometric ratios. Two popular cases are:
- Angle of Elevation: Looking up from the ground to the top of an object (like a tower or tree).
- Angle of Depression: Looking down from a height to a point below (like from a building to the street).
Step-by-Step Illustration
Let’s see how to solve a classic application of trigonometry problem:
Example: A pole is 10 m high. A wire is attached from the top of the pole to a point on the ground, forming an angle of 30° with the ground. Find the length of the wire.
2. Use sine: \( \sin 30^\circ = \frac{10}{\text{length of wire}} \)
3. Substitute values: \( 0.5 = \frac{10}{\text{length of wire}} \)
4. Rearranging: \( \text{length of wire} = \frac{10}{0.5} = 20 \) m
5. Final Answer: The length of the wire is 20 m.
Speed Trick or Vedic Shortcut
Here’s a quick shortcut for students solving trigonometry height and distance problems: Remember “SOH-CAH-TOA” — Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. Before starting, label all sides clearly and mark the angle.
Example Trick: If the problem gives an angle and an adjacent side, use cosine; if opposite side and angle, use sine. This saves time and reduces confusion during exams, especially in one-mark or two-mark questions. Vedantu’s live classes teach you more such time-saving tricks!
Try These Yourself
- You stand 15 m from a tree. The angle of elevation to the top is 60°. Find the height of the tree.
- A ladder leans against a wall making a 45° angle with the ground. If the foot is 5 m from the wall, find the length of the ladder.
- A kite is flying at a height of 40 m. The string makes a 30° angle with the ground. Find the length of the string.
Frequent Errors and Misunderstandings
- Mixing up adjacent and opposite sides based on the angle
- Forgetting to convert units (m to cm, etc.) where needed
- Selecting the wrong trigonometric ratio (using sin when tan is needed)
- Ignoring whether the angle is of elevation or depression
Relation to Other Concepts
The idea of application of trigonometry connects closely with height and distance and introduction to trigonometry. Mastering this helps with understanding advanced geometry, coordinate geometry, and real-world math problems appearing in competitive exams.
Classroom Tip
A quick way to remember which ratio to use: “O is for Opposite — Sine; A is for Adjacent — Cosine; Opposite over Adjacent — Tangent.” Teachers at Vedantu often encourage students to sketch diagrams whenever possible, label the triangle, and write down SOH-CAH-TOA to make the solution process foolproof.
We explored application of trigonometry—from its definition, formulae, step-by-step examples, common mistakes, and connections to other subjects. Continue practicing with Vedantu and you’ll soon gain full confidence in solving all types of height and distance or angle-based word problems!
Internal Links for Extra Learning
- Introduction to Trigonometry
- Height and Distance
- Trigonometric Ratios of Standard Angles
- Trigonometry Questions for Class 10
FAQs on Application of Trigonometry in Real Life and Board Exams
1. What is the application of trigonometry in class 10 Maths?
In Class 10, trigonometry's applications primarily focus on solving height and distance problems. You'll learn to use trigonometric ratios (sin, cos, tan) to calculate unknown sides or angles in right-angled triangles representing real-world scenarios, such as finding the height of a building or the distance across a river, using given angles of elevation or depression.
2. Where do we use trigonometric ratios in real life?
Trigonometric ratios have numerous real-life applications. They are crucial in fields like navigation (calculating distances and bearings), surveying (measuring land areas and heights), engineering (designing structures and machines), astronomy (determining distances to stars and planets), and even in creating computer graphics and video games.
3. How do I solve height and distance problems in trigonometry?
Solving height and distance problems involves these steps: 1. Draw a right-angled triangle representing the scenario. 2. Identify the known and unknown sides (opposite, adjacent, hypotenuse). 3. Choose the appropriate trigonometric ratio (sin, cos, or tan) based on the known and unknown values. 4. Substitute the values into the chosen trigonometric ratio formula and solve for the unknown.
4. What are common questions asked from this topic in board exams?
Board exams often include questions on calculating heights and distances using angles of elevation and depression. Expect problems involving various scenarios like finding the height of a tower, the width of a river, or the distance between two points using trigonometric ratios and right-angled triangles. Practice problems focusing on these scenarios will help you perform well.
5. Can you give an example of a practical application of sin, cos, or tan?
Imagine you need to find the height of a tree. If you measure the distance from the tree (adjacent side) and the angle of elevation from the ground to the treetop (angle), you can use the tangent ratio (tan = opposite/adjacent) to calculate the tree's height (opposite side).
6. Why is the angle of elevation always measured from the horizontal and not the vertical?
The angle of elevation is measured from the horizontal because it consistently defines the angle between the horizontal line of sight and the line of sight to an object above the horizontal. Using the horizontal as a reference provides a standardized and easily understood framework for solving height and distance problems.
7. Can trigonometry be applied if the triangle is not right-angled?
Yes, even for non-right-angled triangles, trigonometry can be used. You can employ the sine rule and cosine rule, which are extensions of the trigonometric ratios applicable to all types of triangles. These rules relate the sides and angles of any triangle, allowing you to solve for unknown quantities.
8. If two observers are at different points, how can we use trigonometry to find the distance between them?
If the distance between two observers and the angle subtended by the line connecting the observers are known, you can use trigonometric functions to find the distance between them. You can make use of the sine rule or cosine rule, depending on the available information.
9. What happens if I use the wrong trigonometric ratio by mistake?
Using the incorrect trigonometric ratio will result in an incorrect answer. Carefully analyze the given information (known and unknown sides and angles) and choose the appropriate ratio (sin, cos, or tan) based on the relationship between those elements. Double-check your work to ensure accuracy.
10. How can technology (apps, calculators) help in real-life trigonometry applications?
Calculators and apps significantly simplify real-life trigonometry applications. They allow quick calculation of trigonometric values, easing the process of solving for unknown sides and angles. Many apps even provide visual aids, making it easier to understand and apply trigonometric principles in various situations.
11. What are some common mistakes students make when solving trigonometry problems?
Common mistakes include selecting the wrong trigonometric function, mixing up opposite and adjacent sides in a right-angled triangle, incorrect use of the calculator, and failing to label diagrams clearly. Careful attention to detail and consistent practice can minimize these errors.
12. How can I improve my problem-solving skills in trigonometry?
Consistent practice is key. Work through numerous problems of varying difficulty levels, focusing on understanding the underlying concepts rather than just memorizing formulas. Drawing clear diagrams and carefully labeling sides and angles will aid problem solving. Seek help when you encounter difficulties, either from teachers or online resources.
