Some Applications of Trigonometry — Mathematics Class 10 Notes (CBSE & HBSE)
Free NCERT Mathematics notes for Some Applications of Trigonometry (Class 10) on Siksha Sarovar, aligned to CBSE and Haryana Board (HBSE). This chapter is broken into 3 topics with clear explanations, formulas, solved examples and board-pattern practice — free to read, no sign-up required.
Board exam focus — Some Applications of Trigonometry (CBSE & HBSE)
Heights and Distances is a very practical chapter. CBSE and HBSE both test finding heights of towers, buildings, and distances using tan, sin, cos. Drawing diagrams is essential.
Angles of Elevation and Depression
Key Terminology
| Term | Definition | Diagram |
|---|---|---|
| Horizontal line | The line parallel to the ground from the observer's eye | — |
| Line of sight | Line from observer's eye to object | Diagonal line |
| Angle of Elevation | Angle from horizontal UP to line of sight (observer looks UP) | ∠ above horizontal |
| Angle of Depression | Angle from horizontal DOWN to line of sight (observer looks DOWN) | ∠ below horizontal |
Important Geometry Fact
When a person looks at an object from a height:
- Angle of depression from observer = Angle of elevation from object
(Alternate interior angles with parallel horizontal lines)
Setting Up Height-and-Distance Problems
Step 1: Draw a clear diagram (rough sketch) Step 2: Label: angle, height (h), distance (d), and unknown Step 3: Identify the right triangle Step 4: Choose the trig ratio: tan = opposite/adjacent (most used!) Step 5: Solve for unknown
Most Used Ratios
- tan θ = height/horizontal distance (when both height and distance involved)
- sin θ = height/hypotenuse (when slant distance involved)
Key Formulas
| Setup | Formula |
|---|---|
| Height of tower, horizontal distance known | h = d × tan θ |
| Horizontal distance, height known | d = h / tan θ |
| Slant distance (ladder, rope) | l = h / sin θ |
ALWAYS draw the diagram first. Without a diagram, you may misidentify the triangle!
Two-Triangle Problems
Problems Involving Two Triangles
Many problems involve two right triangles sharing a common side (the height or distance).
Types:
- Two observers, same object: From two points on the same line, angles of elevation of top of tower are α and β. Find height.
- Same observer, two objects: Observer sees two objects at angles of depression α and β. Find distance between objects.
- Moving observer: Speed problems — as observer moves, angle changes.
Standard Two-Triangle Setups
Setup 1: Observer at ground, tower PQ, two points A and B on same side:
- tan α = h/d₁, tan β = h/d₂
- Find h or d₁-d₂
Setup 2: From top of cliff, objects at sea (angles of depression):
- Height of cliff = h
- Object 1 at angle α, Object 2 at angle β
- Distance between objects = h/tan β - h/tan α = h(1/tan β - 1/tan α)
Worked Problem Type
"The angle of elevation of a cloud from a point h metres above a lake is α, and the angle of depression of its reflection is β. Find the height of the cloud."
Let cloud height above lake = H, point is h m above lake.
- tan α = (H-h)/d ...(1)
- tan β = (H+h)/d ...(2) [reflection is h below lake level]
- Divide: tan β/tan α = (H+h)/(H-h)
- Solve for H
CBSE Note: Two-triangle problems are typically 4-5 marks. Practice drawing two triangles sharing a common vertical or horizontal line.
Special Angle Problems and Mixed Applications
Speed and Changing Angle Problems
"A man walking toward a tower observes its angle of elevation changes from 30° to 60°. Find how much closer he moved."
Let height = h, initial distance = d₁, final distance = d₂.
- tan 30° = h/d₁ → d₁ = h√3
- tan 60° = h/d₂ → d₂ = h/√3
- Distance covered = d₁ - d₂ = h√3 - h/√3 = 2h/√3
If specific distance given, solve for h, then find the distance.
Building/Tree Broken by Wind
"A tree is broken by the wind. The broken part makes 30° with ground, tip touches ground 8 m from base. Find original height."
- Let broken part = l (hypotenuse), standing part = p
- tan 30° = p/8 → p = 8/√3
- cos 30° = 8/l → l = 16/√3
- Original height = p + l = 24/√3 = 8√3 m
Summary of Problem Approach
| Step | Action |
|---|---|
| 1 | Draw diagram with right triangle(s) |
| 2 | Label all knowns (angle, distance, height) |
| 3 | Identify which trig ratio connects knowns and unknown |
| 4 | Write the equation: tanθ=opp/adj etc. |
| 5 | Solve algebraically |
| 6 | Rationalize surds (multiply by √3/√3 etc.) |
| 7 | Write answer with units |
Common Surd Simplifications: 1/√3 = √3/3; √3×√3 = 3; h/√3 = h√3/3 CBSE: Always rationalize denominators containing √2, √3 in final answer.
Frequently asked questions
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Yes — the Some Applications of Trigonometry notes for Mathematics (Class 10) on Siksha Sarovar are completely free to read, with no account required.
Do these notes follow CBSE and HBSE?
Yes. The Some Applications of Trigonometry notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.
What does the Some Applications of Trigonometry chapter cover?
Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Some Applications of Trigonometry.