What is the Use of Trigonometry in Remote Sensing?

S6-SA2-0203

Grade Level:

Class 10

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

Definition

What is it?

Trigonometry helps remote sensing by using angles and distances to measure features on Earth from a distance, usually from satellites or aircraft. It allows us to calculate heights of objects, distances between points, and map areas without actually touching them.

Simple Example

Quick Example

Imagine you are flying a kite and want to know how high it is, but you can't climb up. If you know the length of the kite string (hypotenuse) and the angle the string makes with the ground, you can use trigonometry (like sine function) to find the kite's height (opposite side). Remote sensing does something similar but on a much larger scale with satellites.

Worked Example

Step-by-Step

PROBLEM: A satellite orbits Earth and takes a picture of a tall mobile tower. The satellite is 600 km directly above a point on the ground. From the satellite, the angle of depression to the top of the mobile tower is 30 degrees. The angle of depression to the base of the tower is 35 degrees. Calculate the height of the mobile tower.

STEP 1: Draw a diagram. Let S be the satellite, A be the point on the ground directly below S, T be the top of the tower, and B be the base of the tower. We have two right-angled triangles: S-A-T' (where T' is the horizontal projection of T from S) and S-A-B.
---STEP 2: Find the distance from the satellite to the base of the tower (SB). In right triangle SAB, angle SAB = 90 degrees. Angle ASB = 90 - 35 = 55 degrees. We know SA = 600 km. Using tan(angle ASB) = AB / SA, or tan(35) = AB / 600. So, AB = 600 * tan(35) = 600 * 0.700 = 420 km (approx).
---STEP 3: Find the horizontal distance from the satellite's ground point to the top of the tower. This will be the same AB. (Assuming the tower is vertical and on the same horizontal plane for simplicity in this example).
---STEP 4: Calculate the vertical distance from the satellite to the top of the tower's level. Let h_T be the height from A to T. In triangle S A (point on ground directly below T), angle S (point on ground directly below T) A = 90 degrees. Angle S (top of tower) A = 90 - 30 = 60 degrees. Using tan(30) = (height from S to T's level) / AB. So, height from S to T's level = AB * tan(30) = 420 * 0.577 = 242.34 km (approx).
---STEP 5: The height from A to B is 0 (base on ground). The height from A to T is the tower height. The distance from the satellite to the ground is 600 km. The distance from the satellite to the level of the top of the tower is 600 - Tower Height. The angle of depression to the top of the tower is 30 degrees. The horizontal distance from the satellite's ground projection to the tower is X. So, tan(30) = (600 - Tower Height) / X. Also, tan(35) = 600 / X. From the second equation, X = 600 / tan(35) = 600 / 0.700 = 857.14 km.
---STEP 6: Now use X in the first equation: tan(30) = (600 - Tower Height) / 857.14. So, 0.577 = (600 - Tower Height) / 857.14. This means 600 - Tower Height = 0.577 * 857.14 = 494.67 km.
---STEP 7: Tower Height = 600 - 494.67 = 105.33 km.
ANSWER: The height of the mobile tower is approximately 105.33 km.

Why It Matters

Trigonometry is super important in fields like Space Technology for designing satellite orbits and in AI/ML for processing image data from drones. It helps engineers in urban planning to map cities accurately and helps environmental scientists monitor forest cover or water levels from space. You could be a satellite engineer or a data scientist using these skills!

Common Mistakes

MISTAKE: Confusing angle of elevation with angle of depression. | CORRECTION: Angle of elevation is measured upwards from the horizontal, while angle of depression is measured downwards from the horizontal.

MISTAKE: Mixing up sine, cosine, and tangent ratios (SOH CAH TOA). | CORRECTION: Remember SOH (Sine = Opposite/Hypotenuse), CAH (Cosine = Adjacent/Hypotenuse), TOA (Tangent = Opposite/Adjacent). Always identify the opposite, adjacent, and hypotenuse relative to the angle you are working with.

MISTAKE: Not drawing a clear diagram for remote sensing problems. | CORRECTION: Always draw a neat, labeled diagram showing the satellite, ground object, horizontal line, and all known angles and distances. This makes it much easier to identify the correct trigonometric ratio.

Practice Questions

Try It Yourself

QUESTION: A drone is flying at a height of 100 meters. It observes a car at an angle of depression of 45 degrees. How far is the car horizontally from the point directly below the drone? | ANSWER: 100 meters

QUESTION: A remote sensing aircraft is flying at 2 km altitude. It needs to measure the width of a river. From the aircraft, the angles of depression to the two opposite banks of the river are 60 degrees and 30 degrees respectively (assuming the aircraft is directly over one bank's extended line). Calculate the width of the river. | ANSWER: Approximately 2.31 km

QUESTION: A satellite is 800 km above the Earth. It spots a mountain peak. The angle of depression to the base of the mountain is 40 degrees, and to the peak is 38 degrees. What is the approximate height of the mountain? (Assume the satellite is directly above a point on the ground, and the base and peak are along the same horizontal line from that point). | ANSWER: Approximately 50.8 km

MCQ

Quick Quiz

Which trigonometric ratio would you use to find the height of an object if you know the angle of elevation and the horizontal distance to the object?

Sine

Cosine

Tangent

Secant

The Correct Answer Is:

Tangent (TOA) relates the opposite side (height) to the adjacent side (horizontal distance) with respect to the given angle of elevation. Sine and Cosine involve the hypotenuse.

Real World Connection

In the Real World

ISRO (Indian Space Research Organisation) uses trigonometry extensively in its satellite missions, like Chandrayaan and Mangalyaan. When their satellites orbit Earth or the Moon, they use trigonometric calculations to determine precise positions, measure distances to features on the surface, and create detailed maps for weather forecasting, resource management, and disaster relief.

Key Vocabulary

Key Terms

REMOTE SENSING: Gathering information about an object or area from a distance, without physical contact. | ANGLE OF ELEVATION: The angle measured upwards from a horizontal line to a point above. | ANGLE OF DEPRESSION: The angle measured downwards from a horizontal line to a point below. | SATELLITE: An artificial body placed in orbit around the Earth or another planet to collect information or for communication. | TRIGONOMETRIC RATIOS: Ratios of sides of a right-angled triangle (sine, cosine, tangent).

What's Next

What to Learn Next

Now that you understand how trigonometry helps in remote sensing, you can explore "Applications of Trigonometry in Navigation". This will show you how these same principles are used to guide ships, planes, and even cars using GPS, building directly on your knowledge of angles and distances.