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.

**Let’s discuss in detail**

## 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**