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CBSE · Class 10 · Mathematics

Some Applications of Trigonometry

Standard difficulty · 20 questions · 40 marks

Name: Date: Score: ____ / 40

Show your working for Sections B and C. Answers are on the last page — try every question first!

Section A — Choose the correct answer8 × 1 = 8 marks
1.
From a point on the ground 15 m from the foot of a clock tower, the angle of elevation of its top is 45°. Its height is:
  1. 15 m
  2. 30 m
  3. 15√3 m
  4. 5√3 m
2.
The angle of elevation of the top of a lighthouse 60 m high, from a point on level ground, is 60°. How far is the point from its foot?
  1. 60√3 m
  2. 60 m
  3. 20√3 m
  4. 120 m
3.
A kite is flying at a height of 63 m. Its string makes an angle of 30° with the ground. Taking the string as straight, its length is:
  1. 42√3 m
  2. 63 m
  3. 63√2 m
  4. 126 m
4.
A ladder 18 m long leans against a wall and makes an angle of 30° with the ground. How high up the wall does it reach?
  1. 9 m
  2. 9√2 m
  3. 9√3 m
  4. 18 m
5.
The angle of elevation of an object is measured from:
  1. the horizontal, upwards
  2. the horizontal, downwards
  3. the ground at the object
  4. the vertical, upwards
6.
A vertical pole 21 m tall casts a shadow 21 m long. The angle of elevation of the sun is:
  1. 30°
  2. 45°
  3. 60°
  4. 90°
7.
From the top of a cliff 75 m high, the angle of depression of a boat at sea is 45°. How far is the boat from the foot of the cliff?
  1. 150 m
  2. 75√3 m
  3. 25√3 m
  4. 75 m
8.
Assertion (A): A building is 23 m from a point on level ground, and the angle of elevation of its top from there is 45°. So it is 23 m tall.
Reason (R): sin 30° = 1/2.
  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, but R is not the correct explanation of A.
  3. A is true, but R is false.
  4. A is false, but R is true.
Section B — Short answer8 × 2 = 16 marks
9.
A tree breaks in a storm. The broken part bends so that its top touches the ground 2 m from the foot of the tree, making an angle of 60° with the ground. Find the height of the tree before it broke.
10.
The angles of elevation of the top of a tower from two points on the ground, on the same side and in line with its foot, are 30° and 60°. The points are 26 m apart. Find the height of the tower.
11.
From a window 18 m above the ground, the angle of elevation of the top of a building across the road is 60° and the angle of depression of its foot is 45°. Find the height of the building.
12.
Two poles of equal height stand on either side of a road 64 m wide. From a point on the road between them, the angles of elevation of their tops are 60° and 30°. Find the height of the poles and where the point is.
13.
A flagstaff stands on top of a building 7 m high. From a point on the ground, the angles of elevation of the bottom and the top of the flagstaff are 45° and 60°. Find the length of the flagstaff.
14.
From the top of a tower, a man sees a car moving towards the foot of the tower at a steady speed. The angle of depression of the car changes from 30° to 60° in 16 minutes. How much longer will the car take to reach the tower?
15.
A statue 4 m tall stands on a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60° and that of the top of the pedestal is 45°. Find the height of the pedestal.
16.
The angle of elevation of the top of a tower from a point 45 m from its foot is 30°. By how much must the tower be raised so that the angle of elevation becomes 60°?
Section C — Word problems4 × 4 = 16 marks
17.
An aeroplane flies horizontally at a height of 1,620 m. From a point on the ground, the angle of elevation of the plane changes from 60° to 30° in 30 seconds. Find its speed.
18.
From the top of a lighthouse 80 m above the sea, the angles of depression of two ships on the same side, one behind the other, are 30° and 45°. How far apart are the ships?
19.
Priya stands on a bridge 19 m above a river. The angles of depression of the two banks, on opposite sides, are 30° and 45°. How wide is the river?
20.
The angle of elevation of the top of a building from the foot of a tower is 30°, and that of the top of the tower from the foot of the building is 60°. The tower is 93 m high. How high is the building?
AmbifulWorksheets
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CBSE · Class 10 · Mathematics

Answer key

Some Applications of Trigonometry · AW-10M-HAD-2026-V1 · For parents — fold back or tear off before handing the sheet to your child.

QAnswerWorking
1(a) 15 mheight = 15 × tan 45° = 15 m
2(c) 20√3 mdistance = 60 ÷ tan 60° = 20√3 m
3(d) 126 mlength = 63 ÷ sin 30° = 126 m
4(a) 9 mheight = 18 × sin 30° = 9 m
5(a) the horizontal, upwards—
6(b) 45°tan θ = 21 ÷ 21, so θ = 45°
7(d) 75 mthe angle of depression equals the angle of elevation from the boat: distance = 75 ÷ tan 45°
8(b) Both A and R are true, but R is not the correct explanation of A.A is true: tan 45° = 1, so height = distance = 23. R is true but does not explain A.
9(4 + 2√3) mstanding part = 2 tan 60° = 2√3; broken part = 2/cos 60° = 4; total = 4 + 2√3
1013√3 m (the nearer point is 13 m from the foot)h = x√3 and h = (x + 26)/√3, so 3x = x + 26, x = 13, h = 13√3
11(18 + 18√3) m45° gives the road width = 18; the part above the window = 18 tan 60° = 18√3
1216√3 m; 16 m from the first pole and 48 m from the secondh = x√3 = (64 − x)/√3, so 3x = 64 − x, x = 16
137(√3 − 1) m45° gives the distance = 7; building + flagstaff = 7 tan 60° = 7√3
148 minuteswith height h, the distances are h√3 and h/√3; the car covers 2h/√3 in 16 minutes, and h/√3 is half of that
152(√3 + 1) mpedestal p = distance (45°); p + 4 = p√3, so p = 4/(√3 − 1) = 2(√3 + 1)
1630√3 mnow 45 tan 30° = 15√3; needed 45 tan 60° = 45√3; raise by 45√3 − 15√3 = 30√3
1736√3 m/shorizontal distances 1620/√3 and 1620√3; it flies 1620√3 − 1620/√3 = 1080√3 m in 30 s
1880(√3 − 1) mdistances 80 cot 30° = 80√3 and 80 cot 45° = 80
1919(√3 + 1) m19 cot 30° + 19 cot 45° = 19√3 + 19
2031 mdistance = 93 ÷ tan 60° = 31√3; building = 31√3 × tan 30° = 31