What are Applications of Trigonometry in Distances?

S6-SA2-0034

Grade Level:

Class 10

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

Definition

What is it?

Applications of Trigonometry in Distances involve using trigonometric ratios (sine, cosine, tangent) to find unknown distances or heights that are difficult to measure directly. It helps us calculate lengths by relating angles and sides of right-angled triangles formed in real-world situations.

Simple Example

Quick Example

Imagine you are flying a kite and it gets stuck on a tall tree. If you know the length of the string you've let out (the hypotenuse) and the angle the string makes with the ground, you can use trigonometry to find out exactly how tall the tree is (the opposite side) without climbing it.

Worked Example

Step-by-Step

Let's find the height of a building.
1. A boy stands 30 meters away from the base of a building.
---2. He looks up at the top of the building, and the angle of elevation (angle from his eye level to the top) is 45 degrees.
---3. We can form a right-angled triangle where the distance from the boy to the building is the 'adjacent' side (30m), and the height of the building (from his eye level) is the 'opposite' side.
---4. We know that tan(angle) = Opposite / Adjacent.
---5. So, tan(45 degrees) = Height / 30.
---6. Since tan(45 degrees) = 1, we have 1 = Height / 30.
---7. Therefore, Height = 30 meters.
---8. If the boy's eye level is 1.5 meters from the ground, the total height of the building is 30 + 1.5 = 31.5 meters.
Answer: The total height of the building is 31.5 meters.

Why It Matters

Understanding trigonometry for distances is crucial for careers in engineering, architecture, and even space technology. Engineers use it to design bridges and buildings, while ISRO scientists use it to track satellites and calculate distances in space, making our lives safer and more connected.

Common Mistakes

MISTAKE: Confusing Angle of Elevation with Angle of Depression. | CORRECTION: Angle of Elevation is always measured upwards from the horizontal line, while Angle of Depression is measured downwards from the horizontal line.

MISTAKE: Using the wrong trigonometric ratio (e.g., using sin instead of tan). | CORRECTION: Always identify the known side(s) and the unknown side relative to the given angle (Opposite, Adjacent, Hypotenuse) before choosing sin (O/H), cos (A/H), or tan (O/A).

MISTAKE: Not adding the observer's height when calculating total height. | CORRECTION: If the angle is measured from an observer's eye level, remember to add the observer's height to the calculated height to get the total height of the object.

Practice Questions

Try It Yourself

QUESTION: A ladder 10 meters long leans against a wall. If it makes an angle of 60 degrees with the ground, how high up the wall does it reach? | ANSWER: 10 * sin(60) = 10 * (sqrt(3)/2) = 5 * sqrt(3) meters (approx 8.66 meters)

QUESTION: From the top of a 75-meter high lighthouse, the angle of depression to a ship is 30 degrees. How far is the ship from the base of the lighthouse? | ANSWER: Let distance be 'x'. tan(30) = 75/x => x = 75 / tan(30) = 75 * sqrt(3) meters (approx 129.9 meters)

QUESTION: A kite is flying at a height of 60 meters above the ground. The string attached to the kite is tied to a point on the ground. If the angle of elevation of the string with the ground is 60 degrees, assuming no slack in the string, find the length of the string. | ANSWER: Let string length be 'L'. sin(60) = 60/L => L = 60 / sin(60) = 60 / (sqrt(3)/2) = 120 / sqrt(3) = 40 * sqrt(3) meters (approx 69.28 meters)

MCQ

Quick Quiz

A tower is 100 meters high. The angle of elevation of its top from a point on the ground 100 meters away from its base is:

30 degrees

45 degrees

60 degrees

90 degrees

The Correct Answer Is:

Here, the opposite side (height of tower) is 100m and the adjacent side (distance from base) is 100m. Since tan(angle) = Opposite/Adjacent = 100/100 = 1, the angle must be 45 degrees.

Real World Connection

In the Real World

Surveyors use trigonometry extensively to measure land boundaries and create maps for new housing societies or roads. Even drone operators use similar principles to calculate distances and heights for aerial photography or delivering packages, making services like Zepto or Swiggy more efficient.

Key Vocabulary

Key Terms

Angle of Elevation: The angle measured upwards from the horizontal line to an object above the observer. | Angle of Depression: The angle measured downwards from the horizontal line to an object below the observer. | Hypotenuse: The longest side of a right-angled triangle, opposite the right angle. | Opposite Side: The side across from a given angle in a right-angled triangle. | Adjacent Side: The side next to a given angle in a right-angled triangle, not the hypotenuse.

What's Next

What to Learn Next

Now that you understand how trigonometry helps measure distances, you can explore 'Heights and Distances' problems, which combine these concepts to solve more complex real-world scenarios. This will further strengthen your foundation for physics and engineering applications.