What are Applications of Trigonometry in Heights?

S6-SA2-0033

Grade Level:

Class 10

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

Definition

What is it?

Applications of Trigonometry in Heights means using trigonometric ratios (like sine, cosine, and tangent) to find unknown heights or distances of objects that are difficult to measure directly. It helps us calculate how tall something is or how far away it is, using angles and known distances.

Simple Example

Quick Example

Imagine you are standing far away from a tall temple. You want to know its height but can't climb it. If you know how far you are standing from the temple's base and the angle your eyes make when looking up at its top, you can use trigonometry to find the temple's height. It's like finding a missing piece of a puzzle!

Worked Example

Step-by-Step

Problem: A boy is standing 10 meters away from the base of a lamppost. He observes the angle of elevation to the top of the lamppost to be 45 degrees. What is the height of the lamppost? (Assume the boy's height is negligible).

Step 1: Draw a right-angled triangle. The lamppost is the perpendicular side, the distance from the boy to the lamppost is the base, and the line of sight to the top is the hypotenuse.
---Step 2: Identify the knowns: Distance (Base) = 10 meters, Angle of Elevation = 45 degrees. We need to find the Height (Perpendicular).
---Step 3: Choose the correct trigonometric ratio. We have the Base and need the Perpendicular, so we use Tangent (Tan = Perpendicular / Base).
---Step 4: Write the equation: tan(45 degrees) = Height / 10.
---Step 5: Recall the value of tan(45 degrees), which is 1.
---Step 6: Substitute the value: 1 = Height / 10.
---Step 7: Solve for Height: Height = 1 * 10 = 10 meters.
---Step 8: State the answer clearly.
Answer: The height of the lamppost is 10 meters.

Why It Matters

Understanding trigonometry in heights is crucial for careers like engineering, where you design buildings or bridges, or for pilots who calculate flight paths. It's also vital in space technology, helping ISRO scientists track satellites and rockets, and even in creating realistic 3D environments for video games or architectural visualization.

Common Mistakes

MISTAKE: Confusing angle of elevation with angle of depression. | CORRECTION: Angle of elevation is when you look UP at an object, while angle of depression is when you look DOWN at an object.

MISTAKE: Using the wrong trigonometric ratio (e.g., using sin instead of tan when perpendicular and base are involved). | CORRECTION: Remember SOH CAH TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. Identify which sides you know and which you need to find.

MISTAKE: Not drawing a clear diagram, leading to incorrect identification of sides. | CORRECTION: Always draw a neat, labeled right-angled triangle to represent the problem. This helps in correctly identifying the perpendicular, base, and hypotenuse relative to the given angle.

Practice Questions

Try It Yourself

QUESTION: A ladder is leaning against a wall. The foot of the ladder is 5 meters away from the wall. If the ladder makes an angle of 60 degrees with the ground, what is the height of the wall up to where the ladder touches it? (Use tan(60) = 1.732) | ANSWER: Height = 5 * tan(60) = 5 * 1.732 = 8.66 meters.

QUESTION: From a point 100 meters away from the base of a building, the angle of elevation to the top of the building is 30 degrees. Find the height of the building. (Use tan(30) = 1/sqrt(3) or 0.577) | ANSWER: Height = 100 * tan(30) = 100 * (1/sqrt(3)) = 100/1.732 = 57.7 meters (approx).

QUESTION: A kite is flying at a height of 60 meters above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is 60 degrees. Assuming there is no slack in the string, find the length of the string. (Use sin(60) = sqrt(3)/2 or 0.866) | ANSWER: sin(60) = Height / Length of string => Length of string = Height / sin(60) = 60 / (sqrt(3)/2) = 120 / sqrt(3) = 120 / 1.732 = 69.28 meters (approx).

MCQ

Quick Quiz

If the angle of elevation to the top of a tower from a point on the ground, which is 30 meters away from the foot of the tower, is 30 degrees, what is the height of the tower?

30 * sqrt(3) meters

30 / sqrt(3) meters

30 meters

15 meters

The Correct Answer Is:

We use tan(angle) = Perpendicular/Base. Here, tan(30) = Height/30. Since tan(30) = 1/sqrt(3), we get Height = 30 * (1/sqrt(3)) = 30/sqrt(3) meters.

Real World Connection

In the Real World

City planners and architects use trigonometry to calculate building heights and ensure structures are safe and stable. For example, during the construction of a new metro bridge in Delhi, engineers use these principles to determine the exact height of pillars and the length of support cables, ensuring precise alignment and safety for millions of commuters.

Key Vocabulary

Key Terms

Angle of Elevation: The angle formed when you look upwards from the horizontal line to an object above. | Angle of Depression: The angle formed when you look downwards from the horizontal line to an object below. | Line of Sight: The imaginary line from the observer's eye to the object being viewed. | Trigonometric Ratios: Relationships between the angles and sides of a right-angled triangle (sine, cosine, tangent). | Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.

What's Next

What to Learn Next

Great job understanding how trigonometry helps with heights! Next, you can explore 'Trigonometry in Distances and Navigation'. This will show you how these same ideas are used by pilots and sailors to find locations and map routes, building directly on what you've learned about angles and sides.