Class 10 Maths: Height and Distance Problems with Trigonometry

Maths (Application of Trigonometry Class 10): Height and Distance Problems

In the realm of Class 10 Maths, trigonometry plays a crucial role in solving various real-world problems, including height and distance problems. In this article, we'll explore some fundamental concepts and techniques of applying trigonometry to solve such problems.

Trigonometric Functions

For height and distance problems, we primarily use three trigonometric functions: sine, cosine, and tangent. These functions help us relate the angles and sides of a right triangle.

1. Sine (sin): The ratio of the length of the side opposite the angle to the length of the hypotenuse.
2. Cosine (cos): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
3. Tangent (tan): The ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Solving Height and Distance Problems

To solve height and distance problems, we follow these steps:

1. Set up the problem: Clearly, identify the given information and unknowns.
2. Sketch the diagram: Draw a sketch of the situation based on the information given.
3. Label the angles and sides: Label the angle of interest, as well as the sides opposite and adjacent to the angle.
4. Apply trigonometry: Choose the appropriate trigonometric function for the given problem, and use the given angle to find the unknown side length.
5. Solve for the unknown: Solve the equation to find the value of the unknown side length.

Here are some examples:

Example 1: A ladder is placed against a wall. The ladder is 15 meters long, and it makes an angle of 60° with the ground. How far up the wall does the ladder reach?

1. Set up the problem: The unknown is the height of the wall reached by the ladder.
2. Sketch the diagram: Draw a right triangle with the ladder as the hypotenuse, the ground as the base, and the wall as the side opposite the angle.
3. Label the angles and sides: Label the angle as 60°, the ladder as the hypotenuse (15 m), and the side opposite the angle as the height of the wall.
4. Apply trigonometry: Use the tangent function (tan) to find the height of the wall: tan(60°) = (opposite side) / (adjacent side).
5. Solve for the unknown: tan(60°) = 15 / (adjacent side); 15 / adjacent side = √3; adjacent side = 15 / √3.

Example 2: A 30-meter long cable is suspended between two poles that are 20 meters apart. Find the height of the cable at the midpoint between the poles.

1. Set up the problem: The unknown is the height of the cable at the midpoint between the poles.
2. Sketch the diagram: Draw a right triangle with the cable as the hypotenuse, the distance between the poles as the base, and the side opposite the midpoint angle as the height of the cable at the midpoint.
3. Label the angles and sides: Label the angle as 30°, the distance between the poles as the base (20 m), and the height of the cable at the midpoint as the side opposite the angle.
4. Apply trigonometry: Use the sine function (sin) to find the height of the cable at the midpoint: sin(30°) = (opposite side) / (hypotenuse).
5. Solve for the unknown: sin(30°) = (opposite side) / 30; opposite side = 30 * sin(30°).

By following these steps and using trigonometry, we can solve various height and distance problems involving right triangles. These techniques are crucial for not only understanding mathematical concepts but also for applying trigonometry to real-world scenarios.