Some Applications of Trigonometry Class 10
Download the PDF containing Some Applications of Trigonometry Class 10 notes and a mind map. Revise heights and distances, angles of elevation and depression, and formula selection through clear explanations and worked examples.
How can you calculate a tower’s height without climbing it? How far is a boat from a lighthouse? Class 10 Applications of Trigonometry uses right-angled triangles to answer such questions. A measured angle and a suitable length can help you find a distance that is difficult to measure directly.
These Some Applications of Trigonometry Class 10 notes support your 2026 revision with key definitions, a formula sheet, solved examples and FAQs. The most useful habit is to translate the situation into a labelled diagram before calculating. Once the correct triangle is identified, choosing sine, cosine or tangent becomes much easier.
Heights and Distances Class 10: Important Terms
- Line of sight: The straight line joining the observer’s eye to the point being viewed.
- Angle of elevation: The angle between the horizontal through the observer’s eye and an upward line of sight.
- Angle of depression: The angle between the horizontal through the observer’s eye and a downward line of sight.
- Horizontal distance: The distance measured along a horizontal direction, rather than along the sloping line of sight.
- Vertical difference: The difference in height between the observation point and the point being viewed.
Measure the Angle from the Horizontal
An angle of elevation or depression is measured from a horizontal line, not from a vertical wall. If a given angle is measured from the vertical, its complementary angle is the angle from the horizontal.
Applications of Trigonometry Class 10 Formula Sheet
In a right triangle, let H be the positive vertical difference, D the horizontal distance and L the line-of-sight length. Let θ be the acute angle the line of sight makes with the horizontal.
| Ratio or Quantity | Formula | Use |
|---|---|---|
| Tangent | tan θ = H/D | Height difference and horizontal distance. |
| Sine | sin θ = H/L | Height difference and line of sight. |
| Cosine | cos θ = D/L | Horizontal distance and line of sight. |
| Height difference | H = D tan θ | When D and θ are known. |
| Horizontal distance | D = H/tan θ | When H and θ are known. |
| Line-of-sight length | L = H/sin θ | When H and θ are known. |
Standard Values Used in Heights-and-Distances Questions
| Angle | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
How to Draw the Correct Diagram
- Draw the horizontal ground or reference line.
- Draw vertical objects perpendicular to it.
- Mark the observer’s eye position and any stated eye height.
- Join the eye to the point being viewed to show the line of sight.
- Mark the angle from the correct horizontal line.
- Label known lengths and define the unknown quantity.
A rough diagram does not need to be drawn to scale, but its labels and relationships must match the question. Do not assume a sloping distance is horizontal or use an object’s total height when the triangle contains only a height difference.
Some Applications of Trigonometry Class 10 Questions with Solutions
Example 1: Height of a Tower
Question: From a point on level ground 20 m from the foot of a vertical tower, the angle of elevation of its top is 60°. Find the tower’s height, treating the observation point as being at ground level.
Let the height be h m.
tan 60° = h/20
√3 = h/20
h = 20√3 m
Example 2: Include the Observer’s Eye Height
Question: An observer’s eyes are 1.6 m above level ground. Standing 10 m from a vertical building, the observer sees its top at an elevation of 45°. Find the building’s height.
Let the building’s height be h m.
Vertical difference above eye level = h − 1.6
tan 45° = (h − 1.6)/10
1 = (h − 1.6)/10
h − 1.6 = 10
h = 11.6 m
The triangle gives the height above eye level. Add the eye height to obtain the total height above the same ground level.
Example 3: Angle of Depression from a Lighthouse
Question: The angle of depression of a boat from an observation point 30 m above sea level is 30°. Find the horizontal distance to the boat.
Let the horizontal distance be d m.
The boat’s corresponding angle of elevation is 30°.
tan 30° = 30/d
1/√3 = 30/d
d = 30√3 m
Ladder, Kite and Line-of-Sight Questions
A ladder, taut kite string or line of sight forms the hypotenuse when it slopes between two points. Use sine when the required quantity is the vertical difference, and cosine when it is the horizontal distance.
Example 4: Height Reached by a Ladder
Question: A 10 m ladder rests against a vertical wall and makes an angle of 60° with level ground. Find the height reached on the wall.
sin 60° = h/10
√3/2 = h/10
h = 5√3 m
The ladder is the hypotenuse, so its length belongs in the denominator of the sine ratio.
Two Observation Points: A Worked Example
Example 5: Moving Towards a Tower
Question: The elevation of a tower’s top is 30° from a ground-level observation point. After moving 20 m towards its foot along level ground, the elevation becomes 60°. Find the tower’s height.
Let the nearer distance be x m and the height be h m.
Farther distance = x + 20
tan 60° = h/x
h = x√3
tan 30° = h/(x + 20)
h = (x + 20)/√3
x√3 = (x + 20)/√3
3x = x + 20
2x = 20
x = 10
h = 10√3 m
Check that the nearer point has the larger elevation angle. Both equations must use the same tower height.
Applications of Trigonometry Class 10 Mind Map and Revision 2026
Use the Some Applications of Trigonometry Class 10 mind map to connect line of sight, elevation, depression and right-triangle ratios. Explain each definition aloud, then identify the appropriate ratio in a fresh question.
For 2026 revision, practise simple height questions before moving to eye-height corrections, depression angles and two-point problems. Use the NCERT exercise questions from your textbook edition and supplement them with suitable case-based questions and verified previous year questions with solutions.
Extra Questions for Independent Practice
-
A 12 m pole casts a shadow when the sun’s elevation is 45°.
Find the shadow length on level ground.
Answer: 12 m. -
A tower’s top has elevation 30° from a ground-level point
18√3 m from its foot. Find its height.
Answer: 18 m. -
A taut 40 m kite string makes an angle of 30° with the horizontal.
Find the kite’s vertical height above the hand holding it.
Answer: 20 m. -
A ground-level point is 15 m horizontally from a tower’s foot.
The top has elevation 45°. Find the line-of-sight length.
Answer: 15√2 m.
Check Whether the Answer Is Reasonable
The hypotenuse must be longer than either perpendicular side. At a 45° angle, the vertical difference equals the horizontal distance. For a fixed height, moving closer increases the angle of elevation. These checks help identify diagram and formula errors.
Related Notes and Worksheet Practice
Practise heights and distances using the Some Applications of Trigonometry Class 10 Worksheet . Draw and label each diagram before calculating.
Revise the ratios and standard values with Introduction to Trigonometry Class 10 Notes and Mind Map , or explore the CBSE Class 10 Maths Notes and Mind Maps collection .
Frequently Asked Questions
Does this PDF contain Applications of Trigonometry notes and a mind map?
Yes. The PDF combines notes for understanding the chapter with a mind map for visual revision. Read the explanations first, then use the map to recall definitions and formula choices.
What is the difference between elevation and depression?
An elevation angle is measured upwards from the observer’s horizontal line to the line of sight. A depression angle is measured downwards. Both are measured from the horizontal, not from a vertical object.
Why does an angle of depression equal the corresponding angle of elevation?
The horizontal lines through the two observation levels are parallel. The line of sight acts as a transversal, making the corresponding depression and elevation angles equal as alternate interior angles.
How do I decide whether to use sine, cosine or tangent?
Identify the known and required sides. Use tangent for opposite and adjacent, sine for opposite and hypotenuse, and cosine for adjacent and hypotenuse. Choose the ratio containing the two quantities you need.
Why is tangent used so often in heights-and-distances questions?
Many questions give a horizontal distance and ask for a vertical height difference. Tangent connects these directly without requiring the line-of-sight length. Sine and cosine remain useful when the hypotenuse is given or required.
When should I add the observer’s eye height?
Add it when your calculation gives the height above eye level but the question asks for the total height above the same ground level. If the observer and object stand at different ground levels, include that level difference in the diagram too.
Is the line of sight the same as horizontal distance?
No. The line of sight is the sloping segment from the observer to the viewed point. In the usual right-triangle model, it is the hypotenuse. Horizontal distance is the adjacent side.
How do I handle two observation points on the same side of a tower?
Label the nearer distance x and the farther distance x plus the distance moved. Write one equation for each angle using the same height, then solve the equations together.
What changes when the observation points are on opposite sides?
If the object’s foot lies between the two points on the same straight line, the total separation is the sum of their distances from the foot. Draw the arrangement before deciding whether to add or subtract distances.
How do I solve a tower-on-a-building problem?
Treat the elevation to the building’s top and the elevation to the tower’s top as separate right triangles with the same horizontal distance. Find the two total heights from the reference level, then subtract to obtain the tower’s height.
Should I draw a diagram even if the question does not provide one?
A labelled sketch is strongly useful because it records the angle, reference level and relevant lengths. It also makes the equations easier to explain. Check any specific presentation instructions in the question.
Why might my answer be too small or negative?
Check for a missing eye-height correction, an incorrectly placed angle or a reversed distance relationship. A negative length usually signals an unsuitable equation or interpretation. Keep all measurements in matching units.
Where can I practise Some Applications of Trigonometry Class 10 questions?
Use the Some Applications of Trigonometry Class 10 worksheet after revising the notes and mind map. Begin with direct questions, then practise diagrams involving two observations.
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