Describe how triangulation method is used to measure large distances, by taking suitable example.

The triangulation method is a technique used to measure large distances or survey land by using the principles of trigonometry. It involves creating triangles between two or three known points and then using the angles and distances between those points to calculate the distance to an unknown point.

Let's consider that we want to measure the distance across a river. We'll choose two points A and B on one side of the river, and C on the other side. The baseline will be the distance between points A and B, which can be easily measured using a measuring tape or any other method. Let's assume that the measured baseline AB is 1000 meters.

Now, we need to measure the angles between the points. Using a theodolite or any other angle measuring instrument, we measure the angles at A and B. Let α be the angle at point A between baseline AB and line AC, and β be the angle at point B between baseline AB and line BC. Assume that α is 60 degrees and β is 45 degrees. We also need to calculate the angle γ at point C, which can be found by the fact that the sum of angles in any triangle is equal to 180 degrees. That is: γ = 180 - (α + β) Now calculating angle γ: γ = 180 - (60 + 45) = 180 - 105 = 75 degrees

To calculate the distances to the unknown point C, we can use the law of sines, which states that in any triangle, the ratio of the length of a side and the sine of its opposite angle is the same for all three sides. Using the law of sines, we can write: AC / sin(β) = BC / sin(α) = AB / sin(γ) Now let's calculate the distance AC: AC = AB * sin(β) / sin(γ) = 1000 * sin(45) / sin(75) AC = 1000 * 0.7071 / 0.9659 AC ≈ 731.56 meters Similarly, let's calculate the distance BC: BC = AB * sin(α) / sin(γ) = 1000 * sin(60) / sin(75) BC = 1000 * 0.8660 / 0.9659 BC ≈ 895.68 meters Now, we have successfully measured the large distances AC and BC using the triangulation method. In this example, the distance across the river from point A to point C is approximately 731.56 meters, and from point B to point C is approximately 895.68 meters.