What is the Height of an Object in Trigonometry?

S6-SA2-0108

Grade Level:

Class 10

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

Definition

What is it?

In trigonometry, the 'height' of an object usually refers to its vertical dimension, measured from its base to its top. We use trigonometric ratios like sine, cosine, and tangent to calculate this height indirectly when direct measurement is difficult or impossible.

Simple Example

Quick Example

Imagine you are flying a kite. You know the length of the string you've let out (say, 50 meters) and the angle the string makes with the ground (say, 30 degrees). The 'height' of the kite is how high it is above the ground. You can use trigonometry to find this height without climbing up to the kite!

Worked Example

Step-by-Step

PROBLEM: A ladder is leaning against a wall. The ladder is 10 meters long and makes an angle of 60 degrees with the ground. What is the height of the wall that the ladder reaches?

STEP 1: Identify the knowns and unknowns. We know the hypotenuse (ladder length = 10m) and the angle of elevation (60 degrees). We need to find the opposite side (height of the wall).
---STEP 2: Choose the correct trigonometric ratio. Since we have the hypotenuse and want to find the opposite side, the sine function is appropriate (sin(angle) = Opposite/Hypotenuse).
---STEP 3: Set up the equation: sin(60°) = Height / 10.
---STEP 4: Recall the value of sin(60°). sin(60°) = sqrt(3)/2 or approximately 0.866.
---STEP 5: Substitute the value: 0.866 = Height / 10.
---STEP 6: Solve for Height: Height = 0.866 * 10.
---STEP 7: Calculate the final answer: Height = 8.66 meters.

ANSWER: The ladder reaches a height of 8.66 meters on the wall.

Why It Matters

Calculating object heights using trigonometry is crucial in fields like engineering, architecture, and even space technology. Civil engineers use it to design bridges and buildings, while ISRO scientists use it to track satellite trajectories. It helps professionals build safer structures and explore the universe!

Common Mistakes

MISTAKE: Confusing which side is Opposite, Adjacent, or Hypotenuse. | CORRECTION: Always identify the angle you are working with. The side directly across from it is Opposite, the side next to it (not the hypotenuse) is Adjacent, and the longest side opposite the 90-degree angle is the Hypotenuse.

MISTAKE: Using the wrong trigonometric ratio (e.g., using cosine instead of sine). | CORRECTION: Remember SOH CAH TOA: SOH (Sine = Opposite/Hypotenuse), CAH (Cosine = Adjacent/Hypotenuse), TOA (Tangent = Opposite/Adjacent). Choose the ratio that connects the sides you know and the side you want to find.

MISTAKE: Forgetting to use the correct units or not stating units in the final answer. | CORRECTION: Always ensure your answer includes the correct units (e.g., meters, feet, kilometers) as given in the problem.

Practice Questions

Try It Yourself

QUESTION: A tree casts a shadow 15 meters long when the angle of elevation of the sun is 45 degrees. What is the height of the tree? | ANSWER: 15 meters

QUESTION: From a point on the ground 50 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 approx 0.577) | ANSWER: Approximately 28.85 meters

QUESTION: A boy is standing 20 meters away from a monument. He observes that the angle of elevation to the top of the monument is 60 degrees. If the boy's eye level is 1.5 meters above the ground, what is the total height of the monument? | ANSWER: Approximately 36.14 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?

Sine

Cosine

Tangent

Cotangent

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 adjacent and hypotenuse.

Real World Connection

In the Real World

Surveyors in India use instruments like theodolites to measure angles to distant objects like mountains or tall buildings. By applying trigonometry, they can calculate the exact height of these objects without physically climbing them. This is vital for city planning and construction projects.

Key Vocabulary

Key Terms

ANGLE OF ELEVATION: The angle formed between the horizontal ground and the line of sight to an object above the horizontal. | HYPOTENUSE: The longest side of a right-angled triangle, opposite the right angle. | OPPOSITE SIDE: The side of a right-angled triangle that is across from a given angle. | ADJACENT SIDE: The side of a right-angled triangle that is next to a given angle, but not the hypotenuse. | TRIGONOMETRIC RATIOS: Ratios of the sides of a right-angled triangle, such as sine, cosine, and tangent.

What's Next

What to Learn Next

Now that you understand how to find the height of an object, you can explore 'Angles of Depression' and 'Applications of Trigonometry'. These concepts will show you how to solve even more complex real-world problems involving heights and distances, making you a geometry expert!