Mathematics · Class 10
Applications of Trigonometry
Heights and distances problems using trigonometry.
3 sections15–25 min read6 MCQs · 2 examples
Chapter Overview
Applications of Trigonometry solves real-world height and distance problems using angles of elevation and depression. You model situations with right triangles and apply sin, cos, and tan ratios.
Angles of Elevation and Depression
- Angle of elevation: angle above horizontal from observer's eye
- Angle of depression: angle below horizontal from observer's eye
- Elevation from A to B equals depression from B to A (alternate angles)
- Line of sight connects observer to object
Problem Strategy
- Draw a clear diagram with right triangle
- Mark known side and required side relative to angle
- Choose ratio: tan for opposite/adjacent, sin for opposite/hypotenuse
- Add observer height when problem gives height of eye above ground
Solved Examples
Step-by-step solutions — read each step before checking the final answer.
Example 1: Tower height
From a point 20 m from a tower, angle of elevation to top is 45°. Find height.
- 1tan 45° = h/20
- 21 = h/20
- 3h = 20 m
Answer
Height = 20 m
Example 2: Ladder problem
A ladder 10 m long leans against a wall making 60° with ground. How high does it reach?
- 1Height = opposite to 60°
- 2sin 60° = h/10
- 3h = 10 × √3/2 = 5√3 m
Answer
5√3 m ≈ 8.66 m
Key Points to Remember
- ✓ Always draw diagram before setting up equation
- ✓ Convert mixed units (m and cm) before calculating
- ✓ Depression angle equals elevation from other point
Exam Tips
- • State which trigonometric ratio you use
- • Include observer height in final answer when given
- • Board problems often use 30°, 45°, 60° for clean numbers
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Formula Cheat Sheet
Quick reference — NCERT board exam formulas
Reciprocal & Quotient
- sin θ = 1/cosec θ ; cosec θ = 1/sin θ
- cos θ = 1/sec θ ; sec θ = 1/cos θ
- tan θ = 1/cot θ ; cot θ = 1/tan θ
- tan θ = sin θ/cos θ ; cot θ = cos θ/sin θ
Pythagorean Identities
- sin²θ + cos²θ = 1
- tan²θ + 1 = sec²θ
- cot²θ + 1 = cosec²θ
Angle Addition
- sin(A ± B) = sin A cos B ± cos A sin B
- cos(A ± B) = cos A cos B ∓ sin A sin B
- tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)
Double & Triple Angle
- sin 2θ = 2 sin θ cos θ
- cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1
- tan 2θ = 2 tan θ/(1 − tan²θ)
- sin 3θ = 3 sin θ − 4 sin³θ
- cos 3θ = 4 cos³θ − 3 cos θ
Power Reducing (Integrals)
- sin²θ = (1 − cos 2θ)/2
- cos²θ = (1 + cos 2θ)/2
- tan²θ = (1 − cos 2θ)/(1 + cos 2θ)
Product ↔ Sum
- sin A sin B = ½[cos(A−B) − cos(A+B)]
- cos A cos B = ½[cos(A−B) + cos(A+B)]
- sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2)
- cos A + cos B = 2 cos((A+B)/2) cos((A−B)/2)
Half Angle & Triangle Laws
- sin(θ/2) = ±√[(1 − cos θ)/2]
- Arc length: S = rθ (θ in radians)
- a/sin A = b/sin B = c/sin C (sine rule)
- a² = b² + c² − 2bc cos A (cosine rule)
Practice MCQs — Applications of Trigonometry
Test your understanding with topic-wise multiple choice questions. Explanations appear after each answer.
Question 1 of 6Score: 0/0
Angle of elevation is measured from: