Heights and Distances using Trigonometric Ratios with Solved Examples
Some applications of trigonometry class 10 notes chapter 9 are available with NCERT Solutions at Vedantu. The notes are typically designed by the mathematics masters at the top-notch online education portal keeping in mind the updated pattern and guidelines by the CBSE board. These quick notes on CBSE class 10 maths will help you to significantly refine your trigonometric skills as well get to the core of the topic in no time.
In addition, you can also check other Maths learning resources such as previous year question papers, sample papers, activity quizzes etc. all for free download at Vedantu.
Examples of Class 10 Chapter 9 - Some Applications of Trigonometry
Under this section, you will find the solved questions of chapter 9 – Some Applications of Trigonometry from Class 10 Maths textbook along with answer keys. These solutions are available for free PDF download from the given link at Vedantu official. Let’s get started with the solved Maths Class 10 chapter 9 questions.
Example: A villager is climbing a coconut tree using a 20 m long rope, which is tightly tied from the top of a vertical pole to the ground. Evaluate the height of the coconut tree, if the angle formed by the rope with the ground level is 30°. You can find the illustration below.
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Solution: Given that length of the rope named AC = 20m
Angle formed is ∠ACB = 30°
Let the height AB of the coconut tree be h (meters)
With that, in right ▲ABC,
sin 30° = AB/AC
½= h/20 (since sin 30° = ½)
H = 20/2 = 10metres
Therefore, the height of the coconut tree is 10m
Example: The facilities department of a housing society plans to put two slides for the kids to play in a park. For the kids below the age of 5 years, they want to have a slide whose top is at a height of 1.5 m, and is disposed at an angle of 30° to the ground, while for older children, they prefer to have a steep slide based at a height of 3 m, and inclined at an angle of 60° to the ground. Find out the length of the slide in both the cases?
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Solution:
Let the length of the slide for children below 5years of age x and the length of the slide for older children be y.
Given that: AF = 1.5 m and BC = 3m
∠FEA = 30°and ∠CDB = 60°
In right ▲FAE, sin 30°= AF/EF = 1.5/x
½ = 1.5/x
Thus x= 3m
In right ▲CBD, sin 60°= BC/CD = 3/y
√3/2 = 3/y
Thus, y = 3*2 / √3 = 2√3m
Therefore the length of the slide for children below 5years of age is 3m and the length of the slide for older children is 2√3m.
Example: An air balloon is flying above the ground at a height of 60 m. The string joined to the balloon is temporarily tied to a point on the ground. The inclination of the string with the ground level is 60°. What will be the length of the string, supposing that there is no slack in the string?
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Solution: Given that: AB = 60m and ∠ACB = 60°
Let AC be the length of the string
Then, in the right ▲ABC, sin 30°= AB/AC
= √3/2 = 60/AC
AC= 60*2/√3/ * √3/√3
= 120* √3 / 3
= 40√3
Therefore, the total length of the string is 40√3m
FAQs on Some Applications of Trigonometry for Class 10 Students
1. What are Some Applications of Trigonometry in Class 10?
The main applications of trigonometry in Class 10 are used to find heights and distances using trigonometric ratios. These problems involve angles of elevation and depression and real-life measurements.
- Finding the height of towers, buildings, trees, or mountains
- Calculating the distance of ships or airplanes
- Measuring the width of rivers or roads
- Solving problems using tan θ = opposite / adjacent
2. What is the angle of elevation in trigonometry?
The angle of elevation is the angle formed between the horizontal line and the line of sight when looking at an object above the observer. It is always measured upward from the horizontal.
- Example: If you look at the top of a tower from the ground, the angle formed is the angle of elevation.
- Used in formula: tan θ = height / distance
3. What is the angle of depression?
The angle of depression is the angle formed between the horizontal line and the line of sight when looking at an object below the observer. It is measured downward from the horizontal.
- Example: Looking down from a lighthouse to a boat.
- The angle of depression is equal to the corresponding angle of elevation.
4. What is the formula used in heights and distances problems?
The most commonly used formula in heights and distances is tan θ = opposite / adjacent. This relates height and horizontal distance in a right-angled triangle.
- sin θ = opposite / hypotenuse
- cos θ = adjacent / hypotenuse
- tan θ = opposite / adjacent
5. How do you solve a heights and distances problem in trigonometry?
To solve a heights and distances problem, first form a right-angled triangle and then apply the correct trigonometric ratio. Follow these steps:
- Draw a rough diagram.
- Mark the known sides and angle.
- Choose the appropriate ratio (usually tan θ).
- Substitute values and solve.
6. Why is tan θ mostly used in applications of trigonometry?
The ratio tan θ is mostly used because it directly relates height (opposite) and horizontal distance (adjacent) without involving the hypotenuse. In most real-life problems, we are given ground distance and need to find height.
- tan θ = height / distance
- Easy to apply in angle of elevation problems
7. Can you give a real-life example of trigonometry application?
Yes, trigonometry is used to calculate the height of a building using the angle of elevation and distance from the base. For example:
- Angle of elevation = 30°
- Distance from building = 50 m
- tan 30° = height / 50
8. What are the standard trigonometric values used in Class 10?
The standard trigonometric values for 0°, 30°, 45°, 60°, and 90° are frequently used in Class 10 applications of trigonometry.
- sin 30° = 1/2, cos 30° = √3/2
- sin 45° = 1/√2, cos 45° = 1/√2
- tan 45° = 1
- tan 60° = √3
9. What is the difference between angle of elevation and angle of depression?
The angle of elevation is measured upward from the horizontal, while the angle of depression is measured downward from the horizontal.
- Elevation: Object is above the observer.
- Depression: Object is below the observer.
- Both angles are equal when measured from parallel horizontal lines.
10. What are common mistakes in applications of trigonometry?
Common mistakes in applications of trigonometry include choosing the wrong trigonometric ratio and misinterpreting angles. Students should avoid:
- Confusing angle of elevation with angle of depression
- Using sin or cos instead of tan when height and base are given
- Not drawing a proper diagram
- Forgetting standard trigonometric values
