Trigonometry, the branch of mathematics dealing with the relationships between the angles and sides of triangles, has a wide range of applications across various fields.
Watch Exclusively Curated Playlist for Class 10 Maths Chapter – 9 – Some Applications of Trigonometry
In this video playlist, you will find all the questions with detailed explanations.
Here are 15 questions based on Chapter 9 (“Some Applications of Trigonometry”) from Class 10th NCERT Maths textbook:
Intro Video – Watch Now
- Exercise 9.1, Question 1: – Watch Video
- A tower stands vertically on the ground. From a point on the ground, which is 30 m away from the foot of the tower, the angle of elevation of the top of the tower is 45°. Find the height of the tower.
- Exercise 9.1, Question 2: Watch Video
- A tree breaks due to a storm and the broken part bends so that the top of the tree touches the ground making an angle of 30° with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.
- Exercise 9.1, Question 3: Watch Video
- A ladder leaning against a wall makes an angle of 60° with the ground. The foot of the ladder is 4.6 m away from the wall. Find the length of the ladder.
- Exercise 9.1, Question 4:
- From a point on the ground, the angles of elevation of the bottom and top of a transmission tower fixed at the top of a 20 m high building are 45° and 60°, respectively. Find the height of the transmission tower.
- Exercise 9.1, Question 5:
- Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the tops of the poles are 60° and 30°, respectively. Find the height of the poles and the distance of the point from the poles.
- Exercise 9.1, Question 6:
- 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 60°. Find the height of the tower.
- Exercise 9.1, Question 7:
- The angles of elevation of the top of a tower from two points at distances a and b from the base and in the same straight line with it are complementary. Prove that the height of the tower is √(ab).
- Exercise 9.1, Question 8:
- A kite is flying at a height of 60 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is 60°. Find the length of the string, assuming that there is no slack in the string.
- Exercise 9.1, Question 9:
- A man standing on the deck of a ship, which is 10 m above the water level, observes the angle of elevation of the top of a hill as 60° and the angle of depression of the base of the hill as 30°. Find the distance of the hill from the ship and the height of the hill.
- Exercise 9.1, Question 10:
- Two towers of heights 20 m and 28 m are erected at the same level. A cable is stretched between the tops of the towers. The angle of depression of the foot of the first tower as seen from the top of the second tower is 30°. Find the distance between the two towers.
- Exercise 9.1, Question 11:
- The angles of depression of the top and the bottom of an 8 m tall building from the top of a multi-storeyed building are 30° and 45°, respectively. Find the height of the multi-storeyed building and the distance between the two buildings.
- Exercise 9.1, Question 12:
- The angle of elevation of the top of a hill at a point on the ground is 30°. On walking 50 m towards the hill, the angle of elevation becomes 60°. Find the height of the hill.
- Exercise 9.1, Question 13:
- A balloon is moving vertically upwards. The angle of elevation of the balloon from a point on the ground is 60°. After 2 minutes, the angle of elevation becomes 75°. If the speed of the balloon is 5 m/min, find the height of the balloon when the first observation was made.
- Exercise 9.1, Question 14:
- The top of a 15 m high tower makes an angle of elevation of 30° with the line of sight of the top of a hill. If the distance between the foot of the tower and the hill is 50 m, find the height of the hill.
- Exercise 9.1, Question 15:
- A person standing on the deck of a ship, which is 15 m above water level, observes the angle of elevation of the top of a hill as 60° and the angle of depression of the base of the hill as 30°. Find the distance of the hill from the ship and the height of the hill.
These questions cover a variety of problems related to heights and distances, involving the use of trigonometric ratios to find unknown lengths and angles.
Here are some unique and lesser-known applications of trigonometry:
- Astronomy and Space Exploration:
- Planetary Motion: Trigonometry is essential in calculating the orbits of planets and other celestial bodies. By measuring angles and distances between objects in space, astronomers can predict their positions and movements.
- Distance Measurement: The parallax method, which uses trigonometry, allows astronomers to measure the distance to nearby stars by observing their apparent shift against the background of more distant stars as Earth orbits the Sun.
- Acoustic Engineering:
- Sound Wave Analysis: Trigonometric functions are used to model sound waves and their interactions. This is crucial in designing concert halls and auditoriums to ensure optimal acoustics.
- Noise Cancellation: Trigonometry helps in creating algorithms for active noise cancellation devices, which work by generating sound waves that are out of phase with the noise, effectively canceling it out.
- Medical Imaging:
- CT Scans: Trigonometric algorithms are fundamental in reconstructing images from CT scans. By rotating X-ray sources and detectors around the patient, multiple projections are taken and then combined using trigonometric principles to create a detailed image of the internal structures.
- Ultrasound: Trigonometry is used to interpret the angles and distances of reflected sound waves, helping to create images of organs and tissues inside the body.
- Seismology:
- Earthquake Analysis: Trigonometry helps in determining the epicenter of an earthquake. By measuring the time difference of seismic waves reaching different locations and using triangulation methods, seismologists can pinpoint where the quake originated.
- Wave Propagation: Understanding the propagation of seismic waves through the Earth’s layers involves trigonometric calculations, which are crucial for predicting the impact and strength of earthquakes.
- Robotics and Computer Vision:
- Robot Navigation: Trigonometry is used to calculate angles and distances for navigation and movement. Robots use these calculations to determine their position and plot paths.
- 3D Modeling and Animation: In computer graphics, trigonometry helps create realistic models and animations. Calculations involving angles and distances allow for the rendering of 3D objects and their interactions with light and shadows.
- Civil Engineering:
- Structural Analysis: Trigonometry is used to calculate forces and stresses in various structures such as bridges, buildings, and dams. Understanding the angles and loads helps ensure stability and safety.
- Land Surveying: Surveyors use trigonometric principles to measure large distances and elevations. This data is critical for creating accurate maps and planning construction projects.
- Navigation:
- Marine and Air Navigation: Pilots and sailors use trigonometry to chart their courses. By calculating angles relative to fixed points such as stars, lighthouses, or GPS coordinates, they can navigate accurately even over long distances.
- Satellite Systems: Trigonometry is integral to the functioning of GPS systems. Satellites use trigonometric calculations to determine the precise location of a receiver on Earth.
These applications highlight the versatility and importance of trigonometry in both practical and advanced technological contexts. Its principles are woven into the fabric of numerous scientific and engineering disciplines, underscoring its fundamental role in advancing human knowledge and capability.
