Mathematics Class 10 - Some Applications-Of-Trigonometry Notes

Some Applications of Trigonometry

In this chapter, you will learn how trigonometry is used to solve real-life problems, especially in finding heights and distances. By the end of this chapter, you will be able to use trigonometric ratios to solve practical problems involving right triangles.

Key Concepts

• Trigonometry: The branch of mathematics that deals with the relationships between the sides and angles of triangles.
• Angle of Elevation: The angle formed by the line of sight going up from the horizontal to an object above.
• Angle of Depression: The angle formed by the line of sight going down from the horizontal to an object below.

Applications of Trigonometry

Trigonometry is used to find the heights of buildings, trees, mountains, and distances across rivers or between objects that cannot be measured directly.

• Surveyors use trigonometry to measure land.
• Engineers use it to design buildings and bridges.
• Pilots and sailors use it for navigation.

Solving Height and Distance Problems

To solve these problems, you need to:

• Draw a diagram showing the situation.
• Mark the right triangle, angles, and known values.
• Use trigonometric ratios (sine, cosine, tangent) to find the unknown side or angle.

Trigonometric Ratios

• sin θ = Opposite / Hypotenuse
• cos θ = Adjacent / Hypotenuse
• tan θ = Opposite / Adjacent

Example Problem

Example: A ladder 10 m long leans against a wall. It makes an angle of 60° with the ground. How high does the ladder reach on the wall?

Solution:
Height = 10 × sin 60° = 10 × (√3/2) = 8.66 m

Practice Questions

1. A tree casts a shadow 15 m long when the angle of elevation of the sun is 30°. Find the height of the tree.
2. A man observes the top of a tower at an angle of elevation of 45°. If he is standing 20 m from the base, find the height of the tower.
3. A kite is flying at a height of 50 m. The string makes an angle of 30° with the ground. Find the length of the string.
4. The angle of depression from a lighthouse to a boat is 60°. If the lighthouse is 40 m high, how far is the boat from the base of the lighthouse?
5. Explain the difference between angle of elevation and angle of depression with a diagram.

Challenge Yourself

• Draw and solve a real-life problem involving the angle of elevation and trigonometric ratios.
• Find the height of a building if you know the distance from the building and the angle of elevation to the top.

Did You Know?

• Trigonometry is used in astronomy to calculate distances to stars and planets.
• Ancient Egyptians used simple trigonometry to build the pyramids!

Glossary

• Trigonometry: Study of relationships between angles and sides of triangles.
• Angle of Elevation: Angle above the horizontal line of sight.
• Angle of Depression: Angle below the horizontal line of sight.
• Hypotenuse: The longest side of a right triangle.

Answers to Practice Questions

1. Height = 15 × tan 30° = 15 × (1/√3) ≈ 8.66 m
2. Height = 20 × tan 45° = 20 × 1 = 20 m
3. Length of string = 50 / sin 30° = 50 / 0.5 = 100 m
4. Distance = 40 / tan 60° = 40 / √3 ≈ 23.09 m
5. Angle of elevation is measured upwards from the horizontal; angle of depression is measured downwards from the horizontal.

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Use trigonometry to solve real-world problems and explore the world around you!