9.1. Heights and Distances

In the previous chapter, you have studied about trigonometric ratios. In this chapter, you will be studying about some ways in which trigonometry is used in the life around you.

In this introductory part, the focus is on the application of trigonometric ratios that you learned earlier. It's important to understand how these ratios can help solve real-life problems involving heights and distances. Trigonometry is not just theoretical; it’s a practical tool used in various fields such as architecture, engineering, and even navigation.

Think about how engineers use trigonometry to design buildings and bridges. They need to know not only the height of structures but also how far away to place support beams, and trigonometric ratios provide the answers.

In this figure, the line AC drawn from the eye of the student to the top of the minar is called the line of sight. The student is looking at the top of the minar. The angle BAC, so formed by the line of sight with the horizontal, is called the angle of elevation of the top of the minar from the eye of the student.

The line of sight is the imaginary line extending from an observer's eye to the object they're viewing. The angle of elevation is the angle created when this line of sight goes above the horizontal line of sight. It's crucial for measuring distances and heights because it helps us establish a relationship between the observer's position, the height of the object viewed, and the distances involved.

Imagine you're standing on the ground looking up at a tall building. The angle you tilt your head upwards is the angle of elevation. The steeper the angle, the taller the building appears relative to your position.

In this case, the line of sight is below the horizontal level. The angle formed by the line of sight with the horizontal is called the angle of depression.

The angle of depression occurs when the observer looks downward to see an object. Just like the angle of elevation helps us when looking up, the angle of depression helps determine the height of objects below the observer's line of sight. Understanding both angles is essential for accurately measuring distances and heights in various applications.

Think about a lifeguard on a tall chair at the beach. When they look down at swimmers in the water, they are using an angle of depression to gauge how far away and how deep the water is relative to their position.

To find the height of the minar without actually measuring it, you would need to know the distance from the observer to the base of the minar, the angle of elevation to the top of the minar, and the height of the observer.

To calculate the height of an object such as a minar, you can use the distances involved along with the angles of elevation and depression. By applying trigonometric ratios, you can solve for unknown heights or distances. For example, if you know how far away you are from the minar and the angle at which you look upward to see its top, trigonometric functions like tangent come into play to find the height.

Consider a game of basketball: If a player stands a specific distance from the hoop, by knowing the angle they shoot the ball at, they can estimate how high the hoop is to ensure the ball goes in.

['Example 1: A tower stands vertically on the ground. From a point on the ground, which is 15 m away from the foot of the tower, the angle of elevation of the top of the tower is found to be 60°. Find the height of the tower.', 'Example 2: An electrician has to repair an electric fault on a pole of height 5 m. She needs to reach a point 1.3 m below the top of the pole. What should be the length of the ladder that she should use which, when inclined at an angle of 60° to the horizontal, would enable her to reach the required position?']

These examples illustrate how trigonometric principles help in practical situations. By using the right triangles formed with the objects of interest and the angles of elevation or depression, you can use trigonometric ratios to find unknown heights or distances. There are formulas like tangent (for height over distance) and sine (for hypotenuse calculations) that come into play.

Picture an electrician needing to fix a light on a tall pole: By knowing how high up she needs to go and the angle of her ladder, she can determine both the length of that ladder and how far it needs to be positioned away from the pole.

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