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In this chapter, you will learn about some of the applications of trigonometry in everyday life. Trigonometry is an old science that is being studied by experts all around the world. Arose in the field of astronomy Astronomers have utilised it since then. Calculating distances between the Earth and the planets and stars, for example. In addition to mathematics, trigonometry is employed in geography and navigation. Trigonometry is used to create maps and to calculate the position of an island in respect to longitudes and latitudes. In this chapter, we’ll learn how trigonometry may be used to find the heights and distances of various things without having to measure them.
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Explanation
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Heights and Distances
The line of sight is the line traced from the student's eye to the top of the miner in this diagram. The student is gazing up at the miner’s spire. The angle of elevation of the top of the miner from the student's eye is defined by the angle BAC created by the line of sight with the horizontal.
In this fig. As a result, the line of sight is the line traced from an observer's eye to a point in the thing being seen. When the point being viewed is above the horizontal level, i.e., when we elevate our head to look at the object, the angle created by the line of sight with the horizontal is called the angle of elevation of the point observed. The line of sight is below the horizontal level in this scenario. The angle of depression is the angle produced by the line of sight with the horizontal. The angle formed by the line of sight with the horizontal when the point is below the horizontal level, i.e., when we lower our head to look at the point being observed, is therefore the angle of depression of a point on the object being observed.
You'll need to know the following information:
- the distance DE between the student and the miner’s foot;
- the angle of elevation, BAC, of the miner’s top; and
- the student's height AE.
How can we calculate the height of the miner if the above three conditions are known?
CD = CB + BD in the diagram. Here, BD = AE, which is the student's height. We'll utilise trigonometric ratios of BAC or A to determine BC.
In ABC, the side BC is the polar opposite of the well-known A. Which of the trigonometric ratios are we going to employ now? Which one has the two values we have and the one we need to figure out? Because these ratios require AB and BC, our search narrows down to either tan A or cot A.
As a result, we have tan A = BC /AB or cot A = AB/BC, which when solved gives us BC. The height of the miner may be calculated by adding AE to BC.
Let us now use some problems to demonstrate the technique we just outlined.
Questions and Answers
Ques.: 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. (2 marks)
Solution:
Let's start by drawing a basic diagram to describe the issue (see Fig. 9.4). The tower is represented by AB, the distance between the point and the tower is represented by CB, and the angle of elevation is represented by ACB. We need to figure out the tower's height, which is AB. ACB is also a triangle with a right angle at B. We use the trigonometric ratio tan 60° (or cot 60°) to solve the issue because it includes AB and BC.
Ques.: On a pole with a height of 5 metres, an electrician must fix an electric issue. To complete the repair operation, she must descend 1.3 metres below the top of the pole. What is the length of the ladder she should use that will allow her to reach the appropriate location when angled at a 60° angle to the horizontal? Also, how far from the foot of the pole should she place the foot of the ladder? (You may take 3 = 1.73) (2 marks)
Solution: The electrician must make it to point B on the pole AD. As a result, BD = AD – AB = (5–1.3) m = 3.7 m. The ladder is represented by BC in this case. Its length, i.e., the hypotenuse of the right triangle BDC, must be determined.
Ques. 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°. (2 marks)
Solution:
Ques. 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. (2 marks)
Solution:
Ques. 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 3 m, and inclined at an angle of 60° to the ground. What should be the length of the slide in each case? (2 marks)
Solution:
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.
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