The Hilltop Discovery

In a small village surrounded by mountains, there lived a young boy named Arjun. Arjun loved climbing hills and exploring nature, but he was often frustrated because he couldn’t figure out how tall the distant mountains were. He admired their grandeur and wished he could measure their heights to know just how far he had to climb.

One day, while resting under a large banyan tree, Arjun met his friend, Priya, who was known for her love of mathematics. Seeing his curiosity, Priya decided to help him. "Why don’t we measure the height of that mountain over there?" she suggested, pointing to a tall peak in the distance.

Arjun was excited but confused. "How can we do that? It’s too high to climb!"

Priya smiled. "We don’t need to climb it! We can use some math. Have you heard of triangles?"

Arjun nodded, remembering how triangles were formed by connecting three points. Priya explained, "If we stand a certain distance away from the mountain and look up at its peak, we can create a right triangle. We’ll measure the distance from where we stand to the base of the mountain and the angle at which we look up to the peak."

They found a flat spot about 100 meters from the base of the mountain. Priya took out a protractor and measured the angle of elevation to the peak of the mountain. It turned out to be 30 degrees.

“Now, we can use trigonometry,” she said. “In a right triangle, the height of the mountain is opposite the angle we measured, and the distance we are standing from the base is the adjacent side.”

Priya explained the concept of tangent: "The tangent of an angle is the ratio of the opposite side to the adjacent side. So, if we call the height of the mountain ‘h’, we can write the equation:

\tan(30°) = \frac{h}{100}

Using the known value of , which is  or approximately 0.577, they set up the equation:

\frac{1}{\sqrt{3}} = \frac{h}{100}

Solving for ‘h’, they found that:

h = 100 \times \frac{1}{\sqrt{3}} \approx 57.7 \text{ meters}

Arjun’s eyes widened in amazement. “So, the mountain is about 57.7 meters tall, and we didn’t even have to climb it!”

Priya smiled, proud of their discovery. “That’s the beauty of trigonometry! We can use angles and measurements to find heights and distances without needing to climb or measure directly.”

From that day on, Arjun not only enjoyed climbing hills but also developed a newfound appreciation for math. He realized that trigonometry could help him understand the world around him in a practical way, turning his curiosity into knowledge. As they walked back to the village, he couldn’t wait to share their discovery with others, eager to inspire them to explore the wonders of math and nature together.