In this question, a broken tree forms a right-angled triangle with the ground. The original height of the tree can be found using trigonometry. The distance from the tree’s base to the point where the top touches the ground (8 meters) is the adjacent side of the 30° angle, and the tree’s height is the opposite side.
In a right-angled triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side. Therefore, tan(30°) = (height of the tree)/(8 m).
Given tan(30°) = 1/√3, the equation becomes 1/√3 = (height of the tree)/8.
Solving for the height, we get height of the tree = 8/√3, which simplifies to approximately 4.62 meters. Thus, the original height of the tree is about 4.62 meters.

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Introduction to Trigonometric Problem-Solving

Trigonometry, a fundamental branch of mathematics, is widely used to solve problems involving angles and distances. One common real-world application is in determining the heights of objects, such as trees, poles, or buildings, especially when direct measurement is impractical. The problem presented here involves a tree that has broken and fallen in such a way that it forms a specific angle with the ground. By applying trigonometric principles, we can calculate the original height of the tree. This scenario is an excellent example of how trigonometry can be applied in practical situations, particularly in dealing with the aftermath of natural events like storms.

Understanding the Broken Tree Scenario

In the given problem, a tree breaks during a storm and falls to the ground, forming a 30° angle at the point where the top of the tree touches the ground. The distance from the base of the tree to this point is 8 meters. This situation creates a right-angled triangle, with the tree itself representing one side (opposite to the angle), the distance from the tree’s base to the top as it lies on the ground (8 meters) as the adjacent side, and the original height of the tree as the unknown we seek to determine.

The Role of Tangent in Trigonometry

To solve this problem, we focus on the tangent function in trigonometry. The tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the adjacent side. In our scenario, the tangent function is ideal for finding the height of the tree, as we have the length of the side adjacent to the angle (8 meters) and need to find the length of the side opposite the angle (the tree’s height).

Applying the Tangent Function
Applying the tangent function to the 30° angle, we use the formula tan(30°) = opposite/adjacent. The tangent of 30° is a known value, 1/√3. Substituting the known values into the formula, we get 1/√3 = (height of the tree)/(8 m). This equation allows us to solve for the height of the tree.

Calculating the Height of the Tree

Rearranging the equation to isolate the height of the tree, we multiply both sides by 8 meters. This gives us height of the tree = 8m × 1/√3. Simplifying this equation, we find that the height of the tree is approximately 4.62 meters. This calculation demonstrates how trigonometry can be effectively used to solve problems involving angles and distances in real-world scenarios.

Trigonometry in Practical Applications

This example underscores the practicality of trigonometry in everyday life. The ability to calculate unknown dimensions using angles and distances is invaluable, not just in academic settings but also in real-life situations like assessing damage after natural disasters. Trigonometry bridges the gap between theoretical mathematics and practical applications, highlighting the importance of understanding mathematical concepts for solving real-world problems.

Discuss this question in detail or visit to Class 10 Maths Chapter 9 for all questions.
Questions of 10th Maths Exercise 9.1 in Detail

 A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 30°. A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30° with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree. A contractor plans to install two slides for the children to play in a park. For the children below the age of 5 years, she prefers to have a slide whose top is at a height of 1.5 m, and is inclined at an angle of 30° to the ground, whereas for elder children, she wants to have a steep slide at a height of 3m, and inclined at an angle of 60° to the ground. What should be the length of the slide in each case? The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower is 30°. Find the height of the tower. A kite is flying at a height of 60 m 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°. Find the length of the string, assuming that there is no slack in the string. A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from 30° to 60° as he walks towards the building. Find the distance he walked towards the building. From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a 20 m high building are 45° and 60° respectively. Find the height of the tower. A statue, 1.6 m tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60° and from the same point the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal. The angle of elevation of the top of a building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 50 m high, find the height of the building. Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30°, respectively. Find the height of the poles and the distances of the point from the poles. A TV tower stands vertically on a bank of a canal. From a point on the other bank directly opposite the tower, the angle of elevation of the top of the tower is 60°. From another point 20 m away from this point on the line joining this point to the foot of the tower, the angle of elevation of the top of the tower is 30°. Find the height of the tower and the width of the canal. From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 45°. Determine the height of the tower. As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships. A 1.2 m tall girl spots a balloon moving with the wind in a horizontal line at a height of 88.2 m from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is 60°. After some time, the angle of elevation reduces to 30°. Find the distance travelled by the balloon during the interval. A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be 60°. Find the time taken by the car to reach the foot of the tower from this point.