Some Applications of Trigonometry

How do surveyors find the height of a tower or the width of a river without climbing or crossing it? In this chapter you will apply the trigonometric ratios learnt earlier to such problems of heights and distances. The line of sight is the line drawn from the eye of an observer to the object viewed. When the object is above the horizontal level of the eye, the angle between the line of sight and the horizontal is the angle of elevation; when the object is below, it is the angle of depression. Each problem is modelled by drawing a neat figure with one or more right triangles, marking the known angle and side, and choosing the ratio that links them, usually tan, since height and horizontal distance are the two legs of the right triangle. You will work with angles of 30, 45 and 60 degrees, find heights of towers, buildings and kites, widths of rivers, and distances between ships, and solve two-triangle problems in which an observer moves towards an object.

Detailed Worksheet: Some Applications of Trigonometry Section A - Definitions (10 marks) 1. Define the line of sight with the help of a figure description. (2 marks) 2. Define the angle of elevation and the angle of depression. (2 marks) 3. Why is the angle of elevation of an object from a point P equal to the angle of depression of P from the object? (2 marks) 4. Which trigonometric ratio is most often used in height and distance problems, and why? (2 marks) 5. What happens to the angle of elevation of the top of a tower as an observer walks towards the tower? Explain. (2 marks) Section B - Calculations and Applications (15 marks) 6. A tower stands vertically on the ground. From a point on the ground 30 m away from the foot of the tower, the angle of elevation of its top is 30 degrees. Find the height of the tower. (3 marks) 7. 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 and makes an angle of 60 degrees with the ground. Assuming there is no slack, find the length of the string. (3 marks) 8. A contractor plans to install two slides in a park. For children below 5 years, the slide has its top at a height of 1.5 m and is inclined at 30 degrees to the ground. For elder children, the slide has its top at a height of 3 m and is inclined at 60 degrees. Find the length of each slide. (3 marks) 9. 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 is 30 degrees. (3 marks) 10. The shadow of a tower standing on level ground is found to be 40 m longer when the Sun's altitude is 30 degrees than when it is 60 degrees. Find the height of the tower. (3 marks) Section C - Diagrams (10 marks) 11. Draw a labelled figure showing an observer at the top of a lighthouse looking at a ship at sea. Mark the line of sight, the horizontal line through the eye, and the angle of depression. Explain why the angle of depression equals the angle of elevation of the top of the lighthouse from the ship. (4 marks) 12. A tree breaks due to a storm and the broken part bends so that the top of the tree touches the ground making an angle of 30 degrees with it. The distance between the foot of the tree and the point where the top touches the ground is 8 m. Draw the figure and find the height of the tree. (3 marks) 13. A statue 1.6 m tall stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60 degrees and of the top of the pedestal is 45 degrees. Draw the figure and find the height of the pedestal. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from 30 degrees to 60 degrees as he walks towards the building. Draw the figure and find the distance he walked towards the building. Explain why his height must be subtracted from the building's height. (5 marks) 15. Two poles of equal heights stand on either side of a road 80 m wide. From a point between them on the road, the angles of elevation of the tops of the poles are 60 degrees and 30 degrees. Find the height of the poles and the distances of the point from the poles. Check your answer by verifying that the distances add to 80 m. (5 marks) 16. A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30 degrees, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be 60 degrees. Find the time taken by the car to reach the foot of the tower from this point, and explain why the answer does not depend on the height of the tower. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Draw a neat figure for every question and mark all known angles and lengths. Use sqrt(3) = 1.732 wherever a decimal answer is required.