To find the height of the pole in this scenario, we use trigonometry. The rope forms a right-angled triangle with the pole and the ground. The length of the rope (20 m) is the hypotenuse, and the height of the pole is the opposite side of the angle at the ground. The angle given is 30°.
In a right-angled triangle, the sine of an angle is the ratio of the opposite side to the hypotenuse. Therefore, sin(30°) = (height of the pole)/(length of the rope).
Given that sin(30°)= 1/2, we have 1/2 = (height of the pole)/(20 m).
Solving for the height of the pole, we get height of the pole = 20 m × 1/2 = 10 m.
Thus, the height of the pole is 10 meters.

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## Trigonometric Applications in Real Life

Trigonometry, a branch of mathematics, is not just confined to theoretical problems but has extensive applications in real life. One such application is in determining heights and distances, which is particularly useful in fields like architecture, astronomy, and even in everyday scenarios like the one involving a circus artist and a rope. This scenario presents a practical problem where trigonometry can be applied to find the height of a pole, given the length of a rope and the angle it makes with the ground. Understanding this application requires a basic grasp of trigonometric ratios and their use in right-angled triangles.

### Circus Artist and the Rope

In the given problem, a circus artist is climbing a 20-meter long rope, which forms an angle of 30° with the ground. This setup creates a right-angled triangle, with the rope as the hypotenuse, the pole as one of the sides (opposite to the angle), and the ground as the base (adjacent to the angle). The objective is to find the height of the pole. This type of problem is a classic example of how trigonometry can be used to solve real-world problems involving heights and angles.

#### The Role of Trigonometric Ratios

To solve this problem, we need to understand trigonometric ratios: sine, cosine, and tangent. These ratios relate the angles of a triangle to its sides. In our scenario, the sine function is most applicable. The sine of an angle in a right-angled triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. This relationship is the key to solving our problem.

Applying the Sine Function
For the given problem, we apply the sine function to the 30° angle. The sine of 30° is a known value, which is 1/2. By setting up the equation sin(30°) = opposite/hypotenuse, we can find the height of the pole. Here, the opposite side is the height of the pole, and the hypotenuse is the length of the rope, which is 20 meters. Therefore, the equation becomes 1/2 = height of the pole / 20 meters.

##### Calculating the Height of the Pole

Solving the equation from the previous step, we multiply both sides by 20 meters to isolate the height of the pole. This gives us height of the pole = 20 meters * 1/2. Simplifying this, we find that the height of the pole is 10 meters. This calculation demonstrates how trigonometry can be practically applied to find unknown dimensions in real-life situations, using just an angle and the length of one side of a triangle.

###### The Practicality of Trigonometry

This example illustrates the practicality of trigonometry in everyday life. The ability to calculate unknown heights or distances using angles and lengths is invaluable in many fields. From engineering projects to simple tasks like determining the height of a pole in a circus setup, trigonometry proves to be an essential tool. It bridges the gap between theoretical mathematics and real-world applications, showcasing the importance of mathematical concepts in practical scenarios.

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.