What is Observer's Position in Trigonometry? | Simple Explanation for Class 10

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What is the Concept of an Observer's Position in Trigonometry?

Grade Level:

Class 10

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

Definition

What is it?

In trigonometry, the 'observer's position' refers to the exact location from where someone is looking at an object. This position is crucial because it determines the angle of elevation (looking up) or the angle of depression (looking down) to the object.

Simple Example

Quick Example

Imagine you are standing on the ground looking up at a kite flying high in the sky. Your feet are the observer's position. If you move closer or farther from where the kite is directly above, the angle you have to tilt your head changes, even if the kite's height remains the same.

Worked Example

Step-by-Step

PROBLEM: A boy is standing 10 meters away from the base of a lamppost. He observes the top of the lamppost at an angle of elevation of 45 degrees. How tall is the lamppost?

STEP 1: Identify the observer's position. The boy is standing on the ground, 10 meters away from the lamppost's base.
---STEP 2: Identify the object being observed. The top of the lamppost.
---STEP 3: Draw a right-angled triangle. The distance from the boy to the lamppost base is one side (adjacent), the height of the lamppost is the other side (opposite), and the line of sight to the top of the post is the hypotenuse.
---STEP 4: Use the given angle of elevation (45 degrees) and the adjacent side (10 meters).
---STEP 5: Recall the trigonometric ratio that relates opposite and adjacent sides: tan(angle) = Opposite / Adjacent.
---STEP 6: Substitute the values: tan(45 degrees) = Height / 10 meters.
---STEP 7: We know tan(45 degrees) = 1. So, 1 = Height / 10.
---STEP 8: Solve for Height: Height = 1 * 10 = 10 meters.
ANSWER: The lamppost is 10 meters tall.

Why It Matters

Understanding the observer's position is vital for engineers designing buildings or bridges, as it helps calculate heights and distances accurately. It's also used by pilots for navigation and by architects to plan structures, ensuring safety and stability. Careers like surveying, civil engineering, and even game development rely on this concept.

Common Mistakes

MISTAKE: Confusing the angle of elevation with the angle of depression, or vice-versa. | CORRECTION: Angle of elevation is always measured upwards from the horizontal line of sight, while the angle of depression is measured downwards from the horizontal line of sight.

MISTAKE: Placing the observer's position at the top of a building when they are looking from the ground. | CORRECTION: Always correctly identify where the 'eye' or viewpoint is located. If you're on the ground looking up, your position is on the ground.

MISTAKE: Not considering the observer's height when the problem specifies it. | CORRECTION: If the observer's height is given, remember to add it to or subtract it from the calculated height/distance to get the final answer, especially when the angle is measured from their eye level.

Practice Questions

Try It Yourself

QUESTION: A girl is standing 15 meters from a tree. If the angle of elevation to the top of the tree is 30 degrees, what is the height of the tree? (Assume tan 30 = 1/sqrt(3)) | ANSWER: 15/sqrt(3) meters or 5 * sqrt(3) meters

QUESTION: From the top of a 50-meter tall building, a security guard observes a car on the road at an angle of depression of 60 degrees. How far is the car from the base of the building? (Assume tan 60 = sqrt(3)) | ANSWER: 50/sqrt(3) meters

QUESTION: A 1.5-meter tall boy is standing 20 meters away from a pole. If the angle of elevation to the top of the pole from his eyes is 45 degrees, what is the total height of the pole? | ANSWER: 21.5 meters

MCQ

Quick Quiz

If you are standing on a 10-meter high terrace and look down at a dog on the street, what is your observer's position?

At the dog's location

At the base of the building

At your eye level on the terrace

Halfway between the terrace and the dog

The Correct Answer Is:

The observer's position is where the person (or camera, or sensor) is located. In this case, you are on the terrace, and your eyes are the point from which you are observing, so option C is correct.

Real World Connection

In the Real World

Imagine drone deliveries from companies like Zepto or Swiggy. The drone's camera acts as the observer. Its position and the angle at which it views a delivery spot are used to calculate the exact drop-off location. This technology helps ensure your chai or groceries land exactly where they should!

Key Vocabulary

Key Terms

OBSERVER: The person or device taking the measurement or looking at an object. | ANGLE OF ELEVATION: The angle measured upwards from a horizontal line to an object above. | ANGLE OF DEPRESSION: The angle measured downwards from a horizontal line to an object below. | HORIZONTAL LINE OF SIGHT: An imaginary straight line extending from the observer's eyes, parallel to the ground. | VERTICAL HEIGHT: The straight up-and-down distance of an object.

What's Next

What to Learn Next

Great job understanding the observer's position! Next, you should learn about 'Line of Sight' and 'Horizontal Line' in more detail. These concepts build directly on what you've learned and will help you solve even more complex trigonometry problems involving heights and distances.