Page 1 SOME APPLICATIONS OF TRIGONOMETRY 195 9 9.1 Introduction 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. Trigonometry is one of the most ancient subjects studied by scholars all over the world. As we have said in Chapter 8, trigonometry was invented because its need arose in astronomy. Since then the astronomers have used it, for instance, to calculate distances from the Earth to the planets and stars. Trigonometry is also used in geography and in navigation. The knowledge of trigonometry is used to construct maps, determine the position of an island in relation to the longitudes and latitudes. Surveyors have used trigonometry for centuries. One such large surveying project of the nineteenth century was the ‘Great Trigonometric Survey’ of British India for which the two largest-ever theodolites were built. During the survey in 1852, the highest mountain in the world was discovered. From a distance of over 160 km, the peak was observed from six different stations. In 1856, this peak was named after Sir George Everest, who had commissioned and first used the giant theodolites (see the figure alongside). The theodolites are now on display in the Museum of the Survey of India in Dehradun. SOME APPLICA TIONS OF TRIGONOMETR Y A Theodolite (Surveying instrument, which is based on the Principles of trigonometry, is used for measuring angles with a rotating telescope) 2020-21 Page 2 SOME APPLICATIONS OF TRIGONOMETRY 195 9 9.1 Introduction 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. Trigonometry is one of the most ancient subjects studied by scholars all over the world. As we have said in Chapter 8, trigonometry was invented because its need arose in astronomy. Since then the astronomers have used it, for instance, to calculate distances from the Earth to the planets and stars. Trigonometry is also used in geography and in navigation. The knowledge of trigonometry is used to construct maps, determine the position of an island in relation to the longitudes and latitudes. Surveyors have used trigonometry for centuries. One such large surveying project of the nineteenth century was the ‘Great Trigonometric Survey’ of British India for which the two largest-ever theodolites were built. During the survey in 1852, the highest mountain in the world was discovered. From a distance of over 160 km, the peak was observed from six different stations. In 1856, this peak was named after Sir George Everest, who had commissioned and first used the giant theodolites (see the figure alongside). The theodolites are now on display in the Museum of the Survey of India in Dehradun. SOME APPLICA TIONS OF TRIGONOMETR Y A Theodolite (Surveying instrument, which is based on the Principles of trigonometry, is used for measuring angles with a rotating telescope) 2020-21 196 MATHEMA TICS In this chapter, we will see how trigonometry is used for finding the heights and distances of various objects, without actually measuring them. 9.2 Heights and Distances Let us consider Fig. 8.1 of prvious chapter, which is redrawn below in Fig. 9.1. Fig. 9.1 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. Thus, the line of sight is the line drawn from the eye of an observer to the point in the object viewed by the observer. The angle of elevation of the point viewed is the angle formed by the line of sight with the horizontal when the point being viewed is above the horizontal level, i.e., the case when we raise our head to look at the object (see Fig. 9.2). Fig. 9.2 2020-21 Page 3 SOME APPLICATIONS OF TRIGONOMETRY 195 9 9.1 Introduction 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. Trigonometry is one of the most ancient subjects studied by scholars all over the world. As we have said in Chapter 8, trigonometry was invented because its need arose in astronomy. Since then the astronomers have used it, for instance, to calculate distances from the Earth to the planets and stars. Trigonometry is also used in geography and in navigation. The knowledge of trigonometry is used to construct maps, determine the position of an island in relation to the longitudes and latitudes. Surveyors have used trigonometry for centuries. One such large surveying project of the nineteenth century was the ‘Great Trigonometric Survey’ of British India for which the two largest-ever theodolites were built. During the survey in 1852, the highest mountain in the world was discovered. From a distance of over 160 km, the peak was observed from six different stations. In 1856, this peak was named after Sir George Everest, who had commissioned and first used the giant theodolites (see the figure alongside). The theodolites are now on display in the Museum of the Survey of India in Dehradun. SOME APPLICA TIONS OF TRIGONOMETR Y A Theodolite (Surveying instrument, which is based on the Principles of trigonometry, is used for measuring angles with a rotating telescope) 2020-21 196 MATHEMA TICS In this chapter, we will see how trigonometry is used for finding the heights and distances of various objects, without actually measuring them. 9.2 Heights and Distances Let us consider Fig. 8.1 of prvious chapter, which is redrawn below in Fig. 9.1. Fig. 9.1 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. Thus, the line of sight is the line drawn from the eye of an observer to the point in the object viewed by the observer. The angle of elevation of the point viewed is the angle formed by the line of sight with the horizontal when the point being viewed is above the horizontal level, i.e., the case when we raise our head to look at the object (see Fig. 9.2). Fig. 9.2 2020-21 SOME APPLICATIONS OF TRIGONOMETRY 197 Now, consider the situation given in Fig. 8.2. The girl sitting on the balcony is looking down at a flower pot placed on a stair of the temple. In this case, the line of sight is below the horizontal level. The angle so formed by the line of sight with the horizontal is called the angle of depression. Thus, the angle of depression of a point on the object being viewed is the angle formed by the line of sight with the horizontal when the point is below the horizontal level, i.e., the case when we lower our head to look at the point being viewed (see Fig. 9.3). Fig. 9.3 Now, you may identify the lines of sight, and the angles so formed in Fig. 8.3. Are they angles of elevation or angles of depression? Let us refer to Fig. 9.1 again. If you want to find the height CD of the minar without actually measuring it, what information do you need? Y ou would need to know the following: (i) the distance DE at which the student is standing from the foot of the minar (ii) the angle of elevation, ? BAC, of the top of the minar (iii) the height AE of the student. Assuming that the above three conditions are known, how can we determine the height of the minar? In the figure, CD = CB + BD. Here, BD = AE, which is the height of the student. To find BC, we will use trigonometric ratios of ? BAC or ? A. In ? ABC, the side BC is the opposite side in relation to the known ? A. Now, which of the trigonometric ratios can we use? Which one of them has the two values that we have and the one we need to determine? Our search narrows down to using either tan A or cot A, as these ratios involve AB and BC. 2020-21 Page 4 SOME APPLICATIONS OF TRIGONOMETRY 195 9 9.1 Introduction 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. Trigonometry is one of the most ancient subjects studied by scholars all over the world. As we have said in Chapter 8, trigonometry was invented because its need arose in astronomy. Since then the astronomers have used it, for instance, to calculate distances from the Earth to the planets and stars. Trigonometry is also used in geography and in navigation. The knowledge of trigonometry is used to construct maps, determine the position of an island in relation to the longitudes and latitudes. Surveyors have used trigonometry for centuries. One such large surveying project of the nineteenth century was the ‘Great Trigonometric Survey’ of British India for which the two largest-ever theodolites were built. During the survey in 1852, the highest mountain in the world was discovered. From a distance of over 160 km, the peak was observed from six different stations. In 1856, this peak was named after Sir George Everest, who had commissioned and first used the giant theodolites (see the figure alongside). The theodolites are now on display in the Museum of the Survey of India in Dehradun. SOME APPLICA TIONS OF TRIGONOMETR Y A Theodolite (Surveying instrument, which is based on the Principles of trigonometry, is used for measuring angles with a rotating telescope) 2020-21 196 MATHEMA TICS In this chapter, we will see how trigonometry is used for finding the heights and distances of various objects, without actually measuring them. 9.2 Heights and Distances Let us consider Fig. 8.1 of prvious chapter, which is redrawn below in Fig. 9.1. Fig. 9.1 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. Thus, the line of sight is the line drawn from the eye of an observer to the point in the object viewed by the observer. The angle of elevation of the point viewed is the angle formed by the line of sight with the horizontal when the point being viewed is above the horizontal level, i.e., the case when we raise our head to look at the object (see Fig. 9.2). Fig. 9.2 2020-21 SOME APPLICATIONS OF TRIGONOMETRY 197 Now, consider the situation given in Fig. 8.2. The girl sitting on the balcony is looking down at a flower pot placed on a stair of the temple. In this case, the line of sight is below the horizontal level. The angle so formed by the line of sight with the horizontal is called the angle of depression. Thus, the angle of depression of a point on the object being viewed is the angle formed by the line of sight with the horizontal when the point is below the horizontal level, i.e., the case when we lower our head to look at the point being viewed (see Fig. 9.3). Fig. 9.3 Now, you may identify the lines of sight, and the angles so formed in Fig. 8.3. Are they angles of elevation or angles of depression? Let us refer to Fig. 9.1 again. If you want to find the height CD of the minar without actually measuring it, what information do you need? Y ou would need to know the following: (i) the distance DE at which the student is standing from the foot of the minar (ii) the angle of elevation, ? BAC, of the top of the minar (iii) the height AE of the student. Assuming that the above three conditions are known, how can we determine the height of the minar? In the figure, CD = CB + BD. Here, BD = AE, which is the height of the student. To find BC, we will use trigonometric ratios of ? BAC or ? A. In ? ABC, the side BC is the opposite side in relation to the known ? A. Now, which of the trigonometric ratios can we use? Which one of them has the two values that we have and the one we need to determine? Our search narrows down to using either tan A or cot A, as these ratios involve AB and BC. 2020-21 198 MATHEMA TICS Therefore, tan A = BC AB or cot A = AB , BC which on solving would give us BC. By adding AE to BC, you will get the height of the minar. Now let us explain the process, we have just discussed, by solving some problems. 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. Solution : First let us draw a simple diagram to represent the problem (see Fig. 9.4). Here AB represents the tower, CB is the distance of the point from the tower and ? ACB is the angle of elevation. We need to determine the height of the tower, i.e., AB. Also, ACB is a triangle, right-angled at B. To solve the problem, we choose the trigonometric ratio tan 60° (or cot 60°), as the ratio involves AB and BC. Now, tan 60° = AB BC i.e., 3 = AB 15 i.e., AB = 15 3 Hence, the height of the tower is 15 3 m. Example 2 : An electrician has to repair an electric fault on a pole of height 5 m. She needs to reach a point 1.3m below the top of the pole to undertake the repair work (see Fig. 9.5). 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? Also, how far from the foot of the pole should she place the foot of the ladder? (You may take 3 = 1.73) Fig. 9.4 Fig. 9.5 2020-21 Page 5 SOME APPLICATIONS OF TRIGONOMETRY 195 9 9.1 Introduction 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. Trigonometry is one of the most ancient subjects studied by scholars all over the world. As we have said in Chapter 8, trigonometry was invented because its need arose in astronomy. Since then the astronomers have used it, for instance, to calculate distances from the Earth to the planets and stars. Trigonometry is also used in geography and in navigation. The knowledge of trigonometry is used to construct maps, determine the position of an island in relation to the longitudes and latitudes. Surveyors have used trigonometry for centuries. One such large surveying project of the nineteenth century was the ‘Great Trigonometric Survey’ of British India for which the two largest-ever theodolites were built. During the survey in 1852, the highest mountain in the world was discovered. From a distance of over 160 km, the peak was observed from six different stations. In 1856, this peak was named after Sir George Everest, who had commissioned and first used the giant theodolites (see the figure alongside). The theodolites are now on display in the Museum of the Survey of India in Dehradun. SOME APPLICA TIONS OF TRIGONOMETR Y A Theodolite (Surveying instrument, which is based on the Principles of trigonometry, is used for measuring angles with a rotating telescope) 2020-21 196 MATHEMA TICS In this chapter, we will see how trigonometry is used for finding the heights and distances of various objects, without actually measuring them. 9.2 Heights and Distances Let us consider Fig. 8.1 of prvious chapter, which is redrawn below in Fig. 9.1. Fig. 9.1 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. Thus, the line of sight is the line drawn from the eye of an observer to the point in the object viewed by the observer. The angle of elevation of the point viewed is the angle formed by the line of sight with the horizontal when the point being viewed is above the horizontal level, i.e., the case when we raise our head to look at the object (see Fig. 9.2). Fig. 9.2 2020-21 SOME APPLICATIONS OF TRIGONOMETRY 197 Now, consider the situation given in Fig. 8.2. The girl sitting on the balcony is looking down at a flower pot placed on a stair of the temple. In this case, the line of sight is below the horizontal level. The angle so formed by the line of sight with the horizontal is called the angle of depression. Thus, the angle of depression of a point on the object being viewed is the angle formed by the line of sight with the horizontal when the point is below the horizontal level, i.e., the case when we lower our head to look at the point being viewed (see Fig. 9.3). Fig. 9.3 Now, you may identify the lines of sight, and the angles so formed in Fig. 8.3. Are they angles of elevation or angles of depression? Let us refer to Fig. 9.1 again. If you want to find the height CD of the minar without actually measuring it, what information do you need? Y ou would need to know the following: (i) the distance DE at which the student is standing from the foot of the minar (ii) the angle of elevation, ? BAC, of the top of the minar (iii) the height AE of the student. Assuming that the above three conditions are known, how can we determine the height of the minar? In the figure, CD = CB + BD. Here, BD = AE, which is the height of the student. To find BC, we will use trigonometric ratios of ? BAC or ? A. In ? ABC, the side BC is the opposite side in relation to the known ? A. Now, which of the trigonometric ratios can we use? Which one of them has the two values that we have and the one we need to determine? Our search narrows down to using either tan A or cot A, as these ratios involve AB and BC. 2020-21 198 MATHEMA TICS Therefore, tan A = BC AB or cot A = AB , BC which on solving would give us BC. By adding AE to BC, you will get the height of the minar. Now let us explain the process, we have just discussed, by solving some problems. 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. Solution : First let us draw a simple diagram to represent the problem (see Fig. 9.4). Here AB represents the tower, CB is the distance of the point from the tower and ? ACB is the angle of elevation. We need to determine the height of the tower, i.e., AB. Also, ACB is a triangle, right-angled at B. To solve the problem, we choose the trigonometric ratio tan 60° (or cot 60°), as the ratio involves AB and BC. Now, tan 60° = AB BC i.e., 3 = AB 15 i.e., AB = 15 3 Hence, the height of the tower is 15 3 m. Example 2 : An electrician has to repair an electric fault on a pole of height 5 m. She needs to reach a point 1.3m below the top of the pole to undertake the repair work (see Fig. 9.5). 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? Also, how far from the foot of the pole should she place the foot of the ladder? (You may take 3 = 1.73) Fig. 9.4 Fig. 9.5 2020-21 SOME APPLICATIONS OF TRIGONOMETRY 199 Solution : In Fig. 9.5, the electrician is required to reach the point B on the pole AD. So, BD = AD – AB = (5 – 1.3)m = 3.7 m. Here, BC represents the ladder. We need to find its length, i.e., the hypotenuse of the right triangle BDC. Now, can you think which trigonometic ratio should we consider? It should be sin 60°. So, BD BC = sin 60° or 3.7 BC = 3 2 Therefore, BC = 3.7 2 3 × = 4.28 m (approx.) i.e., the length of the ladder should be 4.28 m. Now, DC BD = cot 60° = 1 3 i.e., DC = 3.7 3 = 2.14 m (approx.) Therefore, she should place the foot of the ladder at a distance of 2.14 m from the pole. Example 3 : An observer 1.5 m tall is 28.5 m away from a chimney. The angle of elevation of the top of the chimney from her eyes is 45°. What is the height of the chimney? Solution : Here, AB is the chimney, CD the observer and ? ADE the angle of elevation (see Fig. 9.6). In this case, ADE is a triangle, right-angled at E and we are required to find the height of the chimney. We have AB = AE + BE = AE + 1.5 and DE = CB = 28.5 m To determine AE, we choose a trigonometric ratio, which involves both AE and DE. Let us choose the tangent of the angle of elevation. Fig. 9.6 2020-21Read More

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