Class 10 › Mathematics Some Applications of Trigonometry Class 10 Mathematics · Chapter 9
Practice all → Summary Questions & answers Practise this chapter → View all Page 1 free about 8 min read , Full kit about 55 min read Page 1 free about 8 min read , Full kit about 55 min read View all 12 amazing Marks Booster kits → This chapter applies trigonometric ratios to practical problems of heights and distances that cannot be measured directly. The key ideas are the angle of elevation (looking up at an object) and the angle of depression (looking down). By modelling a situation as a right-angled triangle and using the appropriate ratio, students can find the height of a building, the width of a river or the distance of a ship. The chapter strengthens the ability to translate a real scene into a diagram and solve it with trigonometry.
Angle of elevation
The angle above the horizontal when looking up at an object.
Angle of depression
The angle below the horizontal when looking down at an object.
Line of sight
The straight line from the observer’s eye to the object.
Heights and distances
Problems solved using right-triangle trigonometry.
Q1 Define the angle of elevation. Answer Q2 Which trigonometric ratio links the opposite side and the hypotenuse? Answer Q3 What is the line of sight? Answer Q4 Define the angle of elevation. Answer Q5 Define the angle of depression. Answer Q6 Which trigonometric ratio links the height and the base in a height-and-distance problem? Answer Q7 A tower casts a shadow such that the angle of elevation of the sun is \(45^\circ\). How are the height and shadow related? Answer Q8 The angle of elevation of the top of a 30 m tower from a point on the ground is \(30^\circ\). Find the distance of the point from the base. Answer Q9 Why does the angle of elevation increase as one moves towards a tower? Answer Q10 What practical problems can be solved using trigonometry? Answer Q1 1. A circus artist is climbing a \(20\,\text{m}\) 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 \(30^\circ\) (see Fig. 9.11). Answer Q2 2. A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle \(30^\circ\) with it. The distance between the foot of the tree to the point where the top touches the ground is \(8\,\text{m}\). Find the height of the tree. Answer Q3 3. 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 \(1.5\,\text{m}\), and is inclined at an angle of \(30^\circ\) to the ground, whereas for elder children, she wants to have a steep slide at a height of \(3\,\text{m}\), and inclined at an angle of \(60^\circ\) to the ground. What should be the length of the slide in each case? Answer Q4 4. The angle of elevation of the top of a tower from a point on the ground, which is \(30\,\text{m}\) away from the foot of the tower, is \(30^\circ\). Find the height of the tower. Answer Q5 5. A kite is flying at a height of \(60\,\text{m}\) 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 \(60^\circ\). Find the length of the string, assuming that there is no slack in the string. Answer Q6 6. A \(1.5\,\text{m}\) tall boy is standing at some distance from a \(30\,\text{m}\) tall building. The angle of elevation from his eyes to the top of the building increases from \(30^\circ\) to \(60^\circ\) as he walks towards the building. Find the distance he walked towards the building. Answer Q7 7. From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a \(20\,\text{m}\) high building are \(45^\circ\) and \(60^\circ\) respectively. Find the height of the tower. Answer Q8 8. A statue, \(1.6\,\text{m}\) 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 \(60^\circ\) and from the same point the angle of elevation of the top of the pedestal is \(45^\circ\). Find the height of the pedestal. Answer Q9 9. The angle of elevation of the top of a building from the foot of the tower is \(30^\circ\) and the angle of elevation of the top of the tower from the foot of the building is \(60^\circ\). If the tower is \(50\,\text{m}\) high, find the height of the building. Answer Q10 10. Two poles of equal heights are standing opposite each other on either side of the road, which is \(80\,\text{m}\) wide. From a point between them on the road, the angles of elevation of the top of the poles are \(60^\circ\) and \(30^\circ\), respectively. Find the height of the poles and the distances of the point from the poles. Answer Q11 11. A TV tower stands vertically on a bank of a canal. From a point on the other bank directly opposite the tower, the angle of elevation of the top of the tower is \(60^\circ\). From another point \(20\,\text{m}\) away from this point on the line joing this point to the foot of the tower, the angle of elevation of the top of the tower is \(30^\circ\) (see Fig. 9.12). Find the height of the tower and the width of the canal. Answer Q12 12. From the top of a \(7\,\text{m}\) high building, the angle of elevation of the top of a cable tower is \(60^\circ\) and the angle of depression of its foot is \(45^\circ\). Determine the height of the tower. Answer Q13 13. As observed from the top of a \(75\,\text{m}\) high lighthouse from the sea-level, the angles of depression of two ships are \(30^\circ\) and \(45^\circ\). If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships. Answer Q14 14. A \(1.2\,\text{m}\) tall girl spots a balloon moving with the wind in a horizontal line at a height of \(88.2\,\text{m}\) from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is \(60^\circ\). After some time, the angle of elevation reduces to \(30^\circ\) (see Fig. 9.13). Find the distance travelled by the balloon during the interval. Answer Q15 15. 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 \(30^\circ\), 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 \(60^\circ\). Find the time taken by the car to reach the foot of the tower from this point. Answer
The angle of elevation is the angle made by the line of sight with the horizontal when the object is:
A at eye level B behind the observer C above the horizontal level D below the horizontal level
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