Angle of depression problems: Class 10 practice
The angle of depression is measured downwards from the horizontal at the observer's eye. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Trigonometric ratios
- Alternate angles
The idea behind angle of depression problems
The angle of depression is measured downwards from the horizontal at the observer's eye. It equals the angle of elevation from the object back to the observer (alternate angles), so use tan = height/horizontal distance.
Step-by-step method
- Draw the horizontal line from the observer.
- Mark the angle of depression and transfer it to the ground.
- Use tan θ = height ÷ distance.
- Solve and round sensibly.
Worked example 1 (Basic)
From the top of a 120 m cliff, the angle of depression of a boat is 60°. How far is the boat from the foot of the cliff?
- tan 60° = 120/d
- d = 120/1.7321 = 69.28 m
- Answer: 69.28 m
Worked example 2 (Standard)
From the top of a 110 m cliff, the angle of depression of a boat is 30°. How far is the boat from the foot of the cliff?
- tan 30° = 110/d
- d = 110/0.5774 = 190.53 m
- Answer: 190.53 m
Worked example 3 (Challenge)
From the top of a 110 m tower, the angles of depression of two cars on the same side are 30° and 60°. How far apart are the cars?
- Near car: 110/tan60° = 63.51 m
- Far car: 110/tan30° = 190.53 m
- Gap = 127.02 m
- Answer: 127.02 m
Common mistakes and misconceptions
- Measuring from the vertical
- Using sin when the slant is unknown
Practice questions
- (Basic) From the top of a 70 m cliff, the angle of depression of a boat is 45°. How far is the boat from the foot of the cliff?
- (Basic) From the top of a 60 m cliff, the angle of depression of a boat is 30°. How far is the boat from the foot of the cliff?
- (Standard) From the top of a 50 m cliff, the angle of depression of a boat is 30°. How far is the boat from the foot of the cliff?
- (Standard) From the top of a 20 m cliff, the angle of depression of a boat is 30°. How far is the boat from the foot of the cliff?
- (Challenge) From the top of a 30 m tower, the angles of depression of two cars on the same side are 30° and 60°. How far apart are the cars?
- (Challenge) From the top of a 60 m tower, the angles of depression of two cars on the same side are 30° and 60°. How far apart are the cars?
Answer key with full working
- Q1: tan 45° = 70/d → d = 70/1.0000 = 70.00 m → Answer: 70.00 m
- Q2: tan 30° = 60/d → d = 60/0.5774 = 103.92 m → Answer: 103.92 m
- Q3: tan 30° = 50/d → d = 50/0.5774 = 86.60 m → Answer: 86.60 m
- Q4: tan 30° = 20/d → d = 20/0.5774 = 34.64 m → Answer: 34.64 m
- Q5: Near car: 30/tan60° = 17.32 m → Far car: 30/tan30° = 51.96 m → Gap = 34.64 m → Answer: 34.64 m
- Q6: Near car: 60/tan60° = 34.64 m → Far car: 60/tan30° = 103.92 m → Gap = 69.28 m → Answer: 69.28 m
Syllabus note
This skill is listed under "Applications of Trigonometry" for Class 10 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is angle of depression problems in?
"Applications of Trigonometry", Class 10 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.