Some Applications Of Trigonometry

CBSE Class 10 — Maths — Previous Year Questions

Q1.VSA1 mark(2020)Easy

The ratio of the length of a vertical rod and the length of its shadow is $1:\sqrt{3}$. Find the angle of elevation of the sun at that moment.

Q2.VSA1 mark(2020)Easy

In Fig. 3, the angles of depressions from the observing positions $O_1$ and $O_2$ respectively of the object $A$ are ________, ________.

Q3.VSA1 mark(2020)Easy

In Figure-3, a tightly stretched rope of length 20 m is tied from the top of a vertical pole to the ground. Find the height of the pole if the angle made by the rope with the ground is $30°$.

Q4.VSA1 mark(2020)Easy

In Figure 4, the angle of elevation of the top of a tower from a point $C$ on the ground, which is $30$ m away from the foot of the tower, is $30°$. Find the height of tower.

Q5.MCQ1 mark(2023)Easy

If a pole 6 m high casts a shadow $2\sqrt{3}$ m long on the ground, then sun's elevation is:

Q6.Case-Study1 mark(2023)Medium

Radio towers are used for transmitting a range of communication services including radio and television. The tower will either act as an antenna itself or support one or more antennas on its structure. On a similar concept, a radio station tower was built in two Sections $A$ and $B$. Tower is supported by wires from a point $O$.

Distance between the base of the tower and point $O$ is $36$ cm. From point $O$, the angle of elevation of the top of the Section $B$ is $30°$ and the angle of elevation of the top of Section $A$ is $45°$.

Based on the above information, answer the following questions:

(i) Find the length of the wire from the point $O$ to the top of Section $B$.

Q7.Case-Study1 mark(2023)Easy

(iii) Find the height of the Section $A$ from the base of the tower.

Q8.MCQ1 mark(2024)Easy

At some time of the day, the length of the shadow of a tower is equal to its height. Then, the Sun's altitude at that time is:

Q9.MCQ1 mark(2025)Easy

A kite is flying at a height of $150$ m from the ground. It is attached to a string inclined at an angle of $30°$ to the horizontal. The length of the string is:

Q10.Case-Study1 mark(2025)Easy

Amrita stood near the base of a lighthouse, gazing up at its towering height. She measured the angle of elevation to the top and found it to be $60°$. Then, she climbed a nearby observation deck, 40 metres higher than her original position and noticed the angle of elevation to the top of lighthouse to be $45°$.





(i) If CD is $h$ metres, find the distance BD in terms of $h$.

Q11.Case-Study1 mark(2025)Easy

(ii) Find distance BC in terms of $h$.

Q12.Assertion-Reason1 mark(2025)Easy

Assertion (A): A ladder leaning against a wall, stands at a horizontal distance of 6 m from the wall. If the height of the wall up to which the ladder reaches is 8 m, then the length of the ladder is 10 m.

Reason (R): The ladder makes an angle of $60°$ with the ground.

Q13.MCQ1 mark(2025)Easy

A 30 m long rope is tightly stretched and tied from the top of pole to the ground. If the rope makes an angle of $60°$ with the ground, the height of the pole is:

Q14.VSA2 marks(2023)Easy

The length of the shadow of a tower on the plane ground is $\sqrt{3}$ times the height of the tower. Find the angle of elevation of the sun.

Q15.VSA2 marks(2023)Easy

The angle of elevation of the top of a tower from a point on the ground which is 30 m away from the foot of the tower, is $30°$. Find the height of the tower.

Q16.VSA2 marks(2023)Easy

Find the length of the shadow on the ground of a pole of height $18$ m when angle of elevation $\theta$ of the sun is such that $\tan \theta =\frac{6}{7}$.

Q17.Case-Study2 marks(2023)Medium

(ii) Find the distance $AB$.

Q18.Case-Study2 marks(2023)Medium

(ii) Find the area of $△OPB$.

Q19.Case-Study2 marks(2025)Medium

(iii) (a) Find the height CE of the lighthouse. (Use $\sqrt{3}=1.73$)

Q20.Case-Study2 marks(2025)Easy

(iii) (b) Find distance AE, if $AC=100$ m.

Q21.SA3 marks(2022)Medium

The angle of elevation of the top of a building from the foot of the tower is $30°$ and the angle of elevation of the top of the tower from the foot of the building is $60°$. If the tower is $50$ m high, then find the height of the building.

Q22.SA3 marks(2022)Medium

From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are $30°$ and $45°$ respectively. If the bridge is at a height of $3$ m from the banks, then find the width of the river.

Q23.SA3 marks(2022)Medium

In Fig. 3, $AB$ is tower of height $50$ m. A man standing on its top, observes two cars on the opposite sides of the tower with angles of depression $30°$ and $45°$ respectively. Find the distance between the two cars.

Q24.SA3 marks(2022)Medium

An aeroplane when flying at a height of 3125 m from the ground passes vertically below another plane at an instant when the angles of elevation of the two planes from the same point on the ground are $30°$ and $60°$ respectively. Find the distance between the two planes at that instant.

Q25.LA4 marks(2020)Hard

A vertical tower stands on a horizontal plane and is surmounted by a vertical flag-staff of height $6$ m. At a point on the plane, the angle of elevation of the bottom and top of the flag-staff are $30°$ and $45°$ respectively. Find the height of the tower. (Take $\sqrt{3}=1.73$)

Q26.LA4 marks(2020)Medium

From a point on the ground, the angles of elevation of the bottom and the top of a tower fixed at the top of a 20 m high building are $45°$ and $60°$ respectively. Find the height of the tower.

Q27.LA4 marks(2020)Hard

If the angle of elevation of a cloud from a point 10 metres above a lake is $30°$ and the angle of depression of its reflection in the lake is $60°$, find the height of the cloud from the surface of lake.

Q28.LA4 marks(2020)Medium

A vertical tower of height 20 m stands on a horizontal plane and is surmounted by a vertical flag-staff of height $h$. At a point on the plane, the angle of elevation of the bottom and top of the flag staff are $45°$ and $60°$ respectively. Find the value of $h$.

Q29.LA4 marks(2020)Medium

A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of $30°$, which is approaching the foot of the tower with a uniform speed. After covering a distance of 50 m, the angle of depression of the car becomes $60°$. Find the height of the tower. (Use $\sqrt{3}=1.73$)

Q30.LA4 marks(2020)Hard

A statue $1.6$ m tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is $60°$ and from the same point the angle of elevation of the top of pedestal is $45°$. Find the height of the pedestal. (Use $\sqrt{3}=1.73$)

Q31.Case-Study4 marks(2022)Hard

Case Study - 2:

Gadisar Lake is located in the Jaisalmer district of Rajasthan. It was built by the King of Jaisalmer and rebuilt by Gadsi Singh in $14^{th}$ century. The lake has many Chhatris. One of them is shown below:



Observe the picture. From a point A $8$ m above from water level, the angle of elevation of top of Chhatri (point B) is $45°$ and angle of depression of its reflection in water (point C) is $60°$. If the height of Chhatri above water level is (approximately) $10$ m, then:

(a) draw a well-labelled figure based on the above information;

(b) find the height ($h$) of the point A above water level.

(Use $\sqrt{3}=1.73$)

Q32.Case-Study4 marks(2022)Hard

Case Study - 1:

**Kite Festival**

Kite festival is celebrated in many countries at different times of the year. In India, every year $14^{th}$ January is celebrated as International Kite Day. On this day many people visit India and participate in the festival by flying various kinds of kites.

The picture given below, shows three kites flying together.



In Fig. 5, the angles of elevation of two kites (Points A and B) from the hands of a man (Point C) are found to be $30°$ and $60°$ respectively. Taking $AD=50$ m and $BE=60$ m, find

(1) the lengths of strings used (take them straight) for kites A and B as shown in the figure.

(2) the distance '$d$' between these two kites.

Q33.LA4 marks(2022)Medium

From the top of an 8 m high building, the angle of elevation of the top of a cable tower is $60°$ and the angle of depression of its foot is $45°$. Determine the height of the tower. (Take $\sqrt{3}=1.732$).

Q34.Case-Study4 marks(2025)Medium

Case Study 2:

A lighthouse stands tall on a cliff by the sea, watching over ships that pass by. One day a ship is seen approaching the shore and from the top of the lighthouse, the angles of depression of the ship are observed to be $30°$ and $45°$ as it moves from point P to point Q. The height of the lighthouse is 50 metres.



Based on the information given above, answer the following questions:
(i) Find the distance of the ship from the base of the lighthouse when it is at point Q, where the angle of depression is $45°$.
(ii) Find the measures of $\angle PBA$ and $\angle QBA$.
(iii) (a) Find the distance travelled by the ship between points P and Q.
**OR**
(b) If the ship continues moving towards the shore and takes 10 minutes to travel from Q to A, calculate the speed of the ship in km/h, from Q to A.

Q35.Case-Study4 marks(2025)Hard

A drone was used to facilitate movement of an ambulance on the straight highway to a point P on the ground where there was an accident. The ambulance was travelling at the speed of 60 km/h. The drone stopped at a point Q, 100 m vertically above the point P. The angle of depression of the ambulance was found to be $30°$ at a particular instant.



Based on above information, answer the following questions:
(i) Represent the above situation with the help of a diagram.
(ii) Find the distance between the ambulance and the site of accident (P) at the particular instant. (Use $\sqrt{3}\approx 1.73$)
(iii) (a) Find the time (in seconds) in which the angle of depression changes from $30°$ to $45°$.

OR

(iii) (b) How long (in seconds) will the ambulance take to reach point P from a point T on the highway such that angle of depression of the ambulance at T is $60°$ from the drone?

Q36.LA5 marks(2023)Hard

The angle of elevation of the top of a tower 24 m high from the foot of another tower in the same plane is $60°$. The angle of elevation of the top of second tower from the foot of the first tower is $30°$. Find the distance between two towers and the height of the other tower. Also, find the length of the wire attached to the tops of both the towers.

Q37.LA5 marks(2023)Hard

A spherical balloon of radius $r$ subtends an angle of $60°$ at the eye of an observer. If the angle of elevation of its centre is $45°$ from the same point, then prove that height of the centre of the balloon is $\sqrt{2}$ times its radius.

Q38.LA5 marks(2023)Medium

As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are $30°$ and $60°$. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships. (Use $\sqrt{3}=1.73$)

Q39.LA5 marks(2023)Medium

From a point on the ground, the angle of elevation of the bottom and top of a transmission tower fixed at the top of 30 m high building are $30°$ and $60°$, respectively. Find the height of the transmission tower. (Use $\sqrt{3}=1.73$)

Q40.LA5 marks(2023)Medium

A straight highway leads to the foot of a tower. A man standing on the top of the 75 m high tower observes two cars at angles of depression of $30°$ and $60°$, which are approaching the foot of the tower. If one car is exactly behind the other on the same side of the tower, find the distance between the two cars. (use $\sqrt{3}=1.73$)

Q41.LA5 marks(2023)Medium

From the top of a 7 m high building, the angle of elevation of the top of a cable tower is $60°$ and the angle of depression of its foot is $30°$. Determine the height of the tower.

Q42.LA5 marks(2023)Hard

One observer estimates the angle of elevation to the basket of a hot air balloon to be $60°$, while another observer $100$ m away estimates the angle of elevation to be $30°$. Find:
(a) The height of the basket from the ground.
(b) The distance of the basket from the first observer's eye.
(c) The horizontal distance of the second observer from the basket.

Q43.LA5 marks(2024)Hard

A pole 6m high is fixed on the top of a tower. The angle of elevation of the top of the pole observed from a point P on the ground is $60°$ and the angle of depression of the point P from the top of the tower is $45°$. Find the height of the tower and the distance of point P from the foot of the tower. (Use $\sqrt{3}=1.73$)

Q44.LA5 marks(2024)Medium

From the top of a $45$ m high light house, the angles of depression of two ships, on the opposite side of it, are observed to be $30°$ and $60°$. If the line joining the ships passes through the foot of the light house, find the distance between the ships. (Use $\sqrt{3}=1.73$)

Q45.LA5 marks(2024)Medium

Two pillars of equal lengths stand on either side of a road which is $100$ m wide, exactly opposite to each other. At a point on the road between the pillars, the angles of elevation of the tops of the pillars are $60°$ and $30°$. Find the length of each pillar and distance of the point on the road from the pillars. (Use $\sqrt{3}=1.732$)

Q46.LA5 marks(2024)Medium

From a point on a bridge across the river, the angles of depression of the banks on opposite sides of the river are $30°$ and $60°$ respectively. If the bridge is at a height of 4 m from the banks, find the width of the river.

Q47.LA5 marks(2025)Hard

Two ships are sailing in the sea on either side of a lighthouse. The angles of depression to the two ships as observed from the top of the lighthouse are $60°$ and $45°$, respectively. If the distance between the ships is $100\left(\frac{1+\sqrt{3}}{\sqrt{3}}\right)$ m, then find the height of the lighthouse.

Q48.LA5 marks(2025)Hard

The angles of depression of the top and the bottom of an 8 m tall building from the top of another multistoried building are $30°$ and $45°$, respectively. Find the height of the multistoried building and the distance between the two buildings.