SEO Title: Applications of Trigonometry Class 10 Notes and Quick Revision Guide
Meta Description: Revise Applications of Trigonometry Class 10 with angle of elevation, angle of depression, tan 30°, tan 45°, tan 60°, solved examples, and quick board exam strategies.
Applications of Trigonometry Class 10 Notes and Quick Revision Guide
Applications of Trigonometry is one of the most important chapters in Class 10 Maths because it connects trigonometric ratios with real-life situations such as heights, distances, towers, poles, shadows, buildings, and line of sight. This chapter is highly scoring in board exams because the question pattern is usually standard, but students often lose marks when they do not draw the diagram correctly or choose the wrong trigonometric ratio.
The chapter becomes much easier when students stop treating it like a memorisation chapter and start treating it like a diagram-based logic chapter. Once the figure is clear, most of the question is already under control. That is why visual understanding matters a lot here.
At Deeksha Vedantu, we always encourage students to solve Applications of Trigonometry step by step. Read the question calmly, identify the angle type, draw the figure neatly, mark the right triangle, and then apply the correct ratio. This simple method makes the chapter far more manageable.
Why Applications of Trigonometry Is Important in Class 10
This chapter is important because it is a regular board-exam area and often appears in 5-mark formats. It also trains students to convert a word problem into a mathematical diagram.
Why Students Should Prepare This Chapter Well
| Reason | Why it matters |
| Board relevance | Questions from this chapter are frequently asked in exams |
| High scoring potential | Standard steps can fetch full marks when done neatly |
| Real-life application | Connects Maths with real situations like heights and distances |
| Strong visual reasoning | Improves diagram interpretation and angle understanding |
| Repeated question style | Similar concepts appear again and again with different wording |
Chapter Overview at a Glance
This quick table helps students revise the chapter before going into detail.
Quick Concept Table
| Topic | Main idea |
| Application of trigonometry | Using trigonometric ratios in real-life height and distance problems |
| Line of sight | Imaginary line from observer’s eye to the object |
| Angle of elevation | Angle below the line of sight when looking upward |
| Angle of depression | Angle above the line of sight when looking downward |
| Most useful ratio | tan θ |
| Main requirement | Correct diagram and correct right triangle |
Most Important Trigonometric Values for This Chapter
In most questions from this chapter, tan values are the most useful because the relationship is usually between perpendicular and base.
Must-Remember tan Values
| Ratio | Value |
| tan 30° | 1/√3 |
| tan 45° | 1 |
| tan 60° | √3 |
Other Useful Values
| Ratio | Value |
| sin 30° | 1/2 |
| sin 45° | 1/√2 |
| sin 60° | √3/2 |
| cos 30° | √3/2 |
| cos 45° | 1/√2 |
| cos 60° | 1/2 |
Why tan Is Used So Often
| Ratio | Formula | Most useful when |
| tan θ | perpendicular/base | Height and horizontal distance are involved |
| sin θ | perpendicular/hypotenuse | Hypotenuse is given or required |
| cos θ | base/hypotenuse | Base and hypotenuse relation is needed |
In most board-level applications of trigonometry questions, tan θ becomes the most convenient ratio.
Angle of Elevation and Angle of Depression
These are the two most important terms in this chapter.
Angle of Elevation
When an observer looks upward at an object, the angle formed between the horizontal line and the line of sight is called the angle of elevation.
Angle of Depression
When an observer looks downward at an object, the angle formed between the horizontal line and the line of sight is called the angle of depression.
Elevation vs Depression Table
| Term | Observer looks | Angle position with respect to line of sight |
| Angle of elevation | Upward | Below the line of sight |
| Angle of depression | Downward | Above the line of sight |
Important Alternate Angle Idea
| Situation | What students should remember |
| Horizontal line at the observer and ground are parallel | Angle of depression is equal to the corresponding angle of elevation |
| A downward line of sight is drawn from a height | The angle at the ground in the triangle often becomes the same by alternate interior angles |
This idea is used in many questions and makes the diagram easier to solve.
What Is the Line of Sight
The line of sight is the straight line from the observer’s eye to the object being seen.
Line of Sight Quick Table
| Term | Meaning |
| Line of sight | Straight line joining observer and object |
| Horizontal line | Straight line parallel to the ground |
| Key use | Helps identify whether the angle is elevation or depression |
The Golden Rule of This Chapter: Draw the Diagram First
Students often lose marks because they rush into formulas without understanding the situation.
Why Diagram Matters So Much
| Without diagram | With diagram |
| Angle gets confused | Angle becomes clear |
| Wrong side may be treated as base or perpendicular | Right triangle becomes easy to identify |
| Ratio selection becomes weak | tan, sin, or cos can be chosen correctly |
| Question feels difficult | Question becomes much easier |
Four-Step Method to Solve Applications of Trigonometry Questions
This is the safest method for Class 10 board preparation.
Solving Method Table
| Step | What to do |
| Step 1 | Read the question carefully and understand the situation |
| Step 2 | Draw the diagram as per the information |
| Step 3 | Search for the right-angled triangle |
| Step 4 | Name the triangle and apply the correct trigonometric ratio |
These four steps make the chapter much easier and reduce careless mistakes.
How to Identify Perpendicular, Base, and Hypotenuse
Students must always identify sides with respect to the given angle.
Side Identification Table
| Side | Meaning in the right triangle |
| Perpendicular | Side opposite to the angle |
| Base | Side adjacent to the angle |
| Hypotenuse | Side opposite to the right angle |
Never label perpendicular and base blindly. Always check the angle first.
Solved Example 1: Car and Tower Problem
The angle of depression of a car standing on the ground from the top of a 75 m high tower is 30°. Find the distance of the car from the base of the tower.
Given
| Quantity | Value |
| Height of tower | 75 m |
| Angle of depression | 30° |
| Distance from base of tower to car | x m |
Step 1: Draw the Situation
Draw the tower vertically on the ground, place the car on the ground, and draw the line of sight from the top of the tower to the car.
Step 2: Use Alternate Angle Idea
Since the horizontal line at the top and the ground are parallel, the angle at the car in the right triangle is also 30°.
Step 3: Identify Sides
For the 30° angle:
- perpendicular = 75 m
- base = x m
Step 4: Apply tan 30°
tan 30° = perpendicular/base
1/√3 = 75/x
x = 75√3
Answer
The distance of the car from the base of the tower is 75√3 m.
Solved Example 2: Tower and Shadow Problem
The shadow of a tower standing on a level plane is found to be 50 m longer when the sun’s elevation is 30° than when it is 60°. Find the height of the tower.
Given
| Quantity | Value |
| Shorter shadow | x m |
| Longer shadow | x + 50 m |
| Height of tower | h m |
| First angle of elevation | 60° |
| Second angle of elevation | 30° |
Step 1: Form the First Triangle
For angle 60°:
tan 60° = h/x
√3 = h/x
h = x√3
Step 2: Form the Second Triangle
For angle 30°:
tan 30° = h/(x + 50)
1/√3 = h/(x + 50)
h = (x + 50)/√3
Step 3: Equate Both Values of h
x√3 = (x + 50)/√3
3x = x + 50
2x = 50
x = 25
Step 4: Find the Height
h = x√3 = 25√3
Answer
The height of the tower is 25√3 m.
Solved Example 3: Two Poles Problem
Two poles are 25 m and 15 m high. The line joining their tops makes an angle of elevation of 45° with the horizontal. Find the distance between the poles.
Given
| Quantity | Value |
| Height of taller pole | 25 m |
| Height of shorter pole | 15 m |
| Angle of elevation | 45° |
| Distance between poles | x m |
Step 1: Find the Vertical Difference
Difference in heights = 25 – 15 = 10 m
Step 2: Form the Right Triangle
The line joining the tops forms a right triangle with:
- perpendicular = 10 m
- base = x m
- angle = 45°
Step 3: Apply tan 45°
tan 45° = perpendicular/base
1 = 10/x
x = 10
Answer
The distance between the two poles is 10 m.
Solved Example 4: Peacock and Snake Problem
A peacock sitting on the top of a tree of height 10 m observes a snake moving on the ground. If the snake is 10√3 m away from the base of the tree, find the angle of depression of the snake from the eye of the peacock.
Given
| Quantity | Value |
| Height of tree | 10 m |
| Distance of snake from base of tree | 10√3 m |
| Angle of depression | θ |
Step 1: Draw the Diagram
Draw the tree vertically, the snake on the ground, and the line of sight from the peacock to the snake.
Step 2: Use Alternate Angle Idea
The angle of depression at the top is equal to the corresponding angle at the base inside the triangle.
Step 3: Identify Sides
For angle θ:
- perpendicular = 10 m
- base = 10√3 m
Step 4: Apply tan θ
tan θ = 10/(10√3)
tan θ = 1/√3
So, θ = 30°
Answer
The angle of depression is 30°.
Quick Logic Behind Most Board Questions
Most questions from this chapter look different in language, but the mathematical structure is often similar.
Common Question Frames
| Question type | What usually changes |
| Tower and car | Angle of depression or elevation |
| Tower and shadow | Two angles and changing shadow length |
| Two buildings or poles | Difference in heights and horizontal distance |
| Tree and bird or peacock | Height, line of sight, and angle |
| Observer moving toward object | Same height with changing distance |
Once students identify the structure, the question becomes much easier.
Most Important Board Patterns from Applications of Trigonometry
Pattern 1: One Right Triangle, One tan Ratio
This is the easiest type. Only one triangle is formed, and one formula is enough.
Pattern 2: Two Triangles, Same Height
This happens in shadow and moving observer questions. Students must form two equations using the same height.
Pattern 3: Difference in Heights
This appears in pole or building questions. First find the height difference, then use tan θ.
Pattern 4: Angle of Depression with Parallel Lines
Students must use the alternate angle idea correctly.
Board Pattern Summary Table
| Pattern | Main skill needed |
| One triangle question | Correct side identification |
| Two triangle question | Equation formation |
| Height-difference question | Visual breakdown |
| Depression question | Alternate angle understanding |
Common Mistakes Students Make in This Chapter
Common Mistakes Table
| Mistake | Correct idea |
| Not reading the question carefully | Always read step by step |
| Drawing a wrong or incomplete diagram | Draw the situation first and neatly |
| Missing the right-angled triangle | Search for the right triangle before using any ratio |
| Choosing the wrong ratio | Identify perpendicular and base with respect to the angle |
| Forgetting alternate angle concept | Depression and elevation often become equal in the figure |
| Forgetting units | Final answer should include metre or the required unit |
Quick Revision Formula Sheet
This section is useful before tests and board exams.
Formula and Value Summary Table
| Item | Formula or value |
| tan 30° | 1/√3 |
| tan 45° | 1 |
| tan 60° | √3 |
| tan θ | perpendicular/base |
| sin θ | perpendicular/hypotenuse |
| cos θ | base/hypotenuse |
| Angle of elevation | Below the line of sight |
| Angle of depression | Above the line of sight |
Final Four Tricks to Remember While Solving
This is the fastest last-minute revision point for students.
Solving Tricks Table
| Trick | What to remember |
| Trick 1 | Read the question carefully |
| Trick 2 | Draw the diagram as per the information |
| Trick 3 | Search out the right-angled triangle |
| Trick 4 | Name the triangle and then apply the ratio |
If students follow these four rules, the chapter becomes much easier.
Practice Questions for Students
Important Practice Set
- The angle of depression of a car standing on the ground from the top of a 60 m tower is 30°. Find the distance of the car from the base of the tower.
- The angle of elevation of the top of a tower from a point on the ground is 45°. If the distance of the point from the base of the tower is 20 m, find the height of the tower.
- A tree casts a shadow of length x m when the sun’s elevation is 60° and a shadow of x + 30 m when the sun’s elevation is 30°. Find the height of the tree.
- Two buildings are 18 m and 8 m high. If the line joining their tops makes an angle of elevation of 45° with the horizontal, find the distance between them.
- A bird sitting on the top of a pole of height 12 m looks at a point on the ground 12√3 m from the base. Find the angle of depression.
FAQs
Q1. What is the most important ratio in Applications of Trigonometry?
In most Class 10 board questions, tan θ is the most important ratio because the relation is usually between perpendicular and base.
Q2. What is angle of elevation?
Angle of elevation is the angle formed when an observer looks upward at an object. It is measured below the line of sight.
Q3. What is angle of depression?
Angle of depression is the angle formed when an observer looks downward at an object. It is measured above the line of sight.
Q4. Why is drawing the diagram important in this chapter?
The diagram helps students identify the right-angled triangle, the correct angle, and the correct trigonometric ratio.
Q5. Why do students make mistakes in Applications of Trigonometry?
Students usually make mistakes because they read the question too fast, draw the figure incorrectly, or choose the wrong ratio.
Q6. How do I know whether to use tan, sin, or cos?
Check which sides are involved with respect to the angle. Use tan for perpendicular and base, sin for perpendicular and hypotenuse, and cos for base and hypotenuse.
Q7. What is the role of alternate angles in this chapter?
In many elevation and depression questions, alternate interior angles help students convert the angle at the top into the angle inside the right triangle on the ground.
Q8. How can I score full marks in this chapter?
Read the question carefully, draw the diagram neatly, find the right triangle, choose the correct ratio, and write the steps clearly with units.
Conclusion
Applications of Trigonometry is one of the most scoring chapters in Class 10 Maths when students understand the logic of diagrams and right triangles. The chapter may look difficult because of the language of the questions, but the actual solution pattern is often simple once the diagram is correct.
The best way to prepare this chapter is to remember the tan values, understand angle of elevation and angle of depression clearly, and practise the standard question types repeatedly. At Deeksha Vedantu, we always remind students that in this chapter, clean diagrams and calm thinking are often more important than speed. Once those two habits are built, the chapter becomes much easier to revise and solve.
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