What is the Use of Trigonometry in Surveying?

S6-SA2-0089

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

Trigonometry helps surveyors measure distances and heights of objects indirectly, especially when direct measurement is difficult or impossible. It uses relationships between angles and sides of triangles to find unknown values like the height of a tall building or the distance across a river.

Simple Example

Quick Example

Imagine you want to know the height of a very tall mango tree in your backyard, but you cannot climb it with a measuring tape. If you stand some distance away, measure the angle from the ground to the top of the tree, and know your distance from the tree's base, trigonometry can help you find its height easily.

Worked Example

Step-by-Step

Problem: A surveyor stands 50 meters away from the base of a cell tower. The angle of elevation to the top of the tower is 30 degrees. What is the height of the cell tower?

Step 1: Draw a right-angled triangle. The distance from the surveyor to the tower is the base (adjacent side = 50 m).
---Step 2: The height of the tower is the opposite side (let's call it 'h').
---Step 3: The angle of elevation is 30 degrees.
---Step 4: We know the adjacent side and want to find the opposite side, so we use the tangent function: tan(angle) = opposite / adjacent.
---Step 5: tan(30 degrees) = h / 50.
---Step 6: We know tan(30 degrees) is approximately 1/sqrt(3) or 0.577.
---Step 7: So, 0.577 = h / 50.
---Step 8: h = 0.577 * 50 = 28.85 meters.
Answer: The height of the cell tower is approximately 28.85 meters.

Why It Matters

Trigonometry is super important in fields like civil engineering to design bridges and buildings, and in space technology for rocket trajectories. Surveyors, architects, and city planners use it daily to map land, plan construction, and ensure everything is built correctly. It's the backbone of many modern developments around us.

Common Mistakes

MISTAKE: Confusing angles of elevation and depression. | CORRECTION: Angle of elevation is measured upwards from the horizontal line. Angle of depression is measured downwards from the horizontal line.

MISTAKE: Using the wrong trigonometric ratio (sine, cosine, tangent) for the given sides and angle. | CORRECTION: Remember SOH CAH TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. Choose based on what you know and what you need to find.

MISTAKE: Not drawing a clear diagram of the problem. | CORRECTION: Always draw a neat, labeled right-angled triangle to visualize the problem. This helps identify the knowns and unknowns correctly.

Practice Questions

Try It Yourself

QUESTION: A ladder leaning against a wall makes an angle of 60 degrees with the ground. If the base of the ladder is 4 meters from the wall, how long is the ladder? | ANSWER: Approximately 8 meters (using cos 60 = adjacent/hypotenuse)

QUESTION: From the top of a 75-meter high lighthouse, the angle of depression to a boat is 45 degrees. How far is the boat from the base of the lighthouse? | ANSWER: 75 meters (using tan 45 = opposite/adjacent)

QUESTION: An airplane is flying at an altitude of 3000 meters. The angle of depression from the plane to a village is 20 degrees. How far is the village from a point directly below the plane on the ground? (Use tan 20 = 0.364) | ANSWER: Approximately 8241.76 meters

MCQ

Quick Quiz

Which trigonometric ratio would you use to find the height of a pole if you know the distance from its base and the angle of elevation to its top?

Sine

Cosine

Tangent

Secant

The Correct Answer Is:

Tangent relates the opposite side (height) to the adjacent side (distance from base), which are the two quantities involved in this scenario. Sine uses hypotenuse, and cosine uses hypotenuse.

Real World Connection

In the Real World

Surveyors in India use instruments called 'Theodolites' or 'Total Stations' which have built-in trigonometric functions. They measure angles and then use trigonometry to calculate distances and elevations for building new roads, railway lines, or even planning new housing colonies in your city. This helps ensure precise measurements for large-scale infrastructure projects like the Delhi Metro or new expressways.

Key Vocabulary

Key Terms

Angle of Elevation: Angle measured upwards from a horizontal line | Angle of Depression: Angle measured downwards from a horizontal line | Theodolite: An optical instrument used by surveyors to measure horizontal and vertical angles | Surveying: The science of determining the terrestrial or three-dimensional position of points and the distances and angles between them | Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.

What's Next

What to Learn Next

Now that you understand how trigonometry helps in surveying, you can explore 'Applications of Trigonometry in Physics'. This will show you how these concepts are used to understand forces, motion, and waves, building on your knowledge of angles and distances in a dynamic way.