Online Coaching for SSC Exams

Size: px

Start display at page:

Download "Online Coaching for SSC Exams"

Daisy Johnson
5 years ago
Views:

1 Online Coaching for SSC Exams Page 1

Trigonometry Subject : Numerical Aptitude Chapter: Trigonometry In the triangles, particularly in right angle triangles, let us take some example from our surrounding where right triangles can be

Now, if a student is looking at the top of the Minar, a right triangle can be imagined to be made, as shown in figure below.

2 Trigonometry Subject : Numerical Aptitude Chapter: Trigonometry In the triangles, particularly in right angle triangles, let us take some example from our surrounding where right triangles can be imagined to be formed. For example:- 1. Suppose the students of a school are visiting Qutub Minar. Now, if a student is looking at the top of the Minar, a right triangle can be imagined to be made, as shown in figure below. Can the student find out the height of the Minar, without actually measuring it? 2. Suppose a girl is sitting on the balcony of her house located on the bank of a river. She is looking down at a flower pot placed on a stair of a temple situated nearby on the other bank of the river. A right triangle is imagined to be made in this situation as shown in Figure below. If you know the height at which the person is sitting, can you find the width of the river? 3. Suppose a hot air balloon is flying in the air. A girl happens to spot the balloon in the sky and runs to her mother to tell her about it. Her mother rushes out of the house to look at the balloon. Now when the girl had spotted the balloon initially it was at point A. When both the mother and daughter came out to see it, it had already traveled to another point B. Can you find the altitude of B from the ground? Page 2

In all the situations given above, the distances or heights can be found by using some mathematical techniques, which come under a branch of mathematics called trigonometry.

In fact, trigonometry is the study of relationships between the sides and angles of a triangle. The earliest known work on trigonometry was recorded in Egypt and Babylon.

3 In all the situations given above, the distances or heights can be found by using some mathematical techniques, which come under a branch of mathematics called trigonometry. The word trigonometry is derived from the Greek words tri (meaning three), gon (meaning sides) and metron (meaning measure). In fact, trigonometry is the study of relationships between the sides and angles of a triangle. The earliest known work on trigonometry was recorded in Egypt and Babylon. Early astronomers used it to find out the distances of the stars and planets from the Earth. Even today, most of the technologically advanced methods used in Engineering and Physical Sciences are based on trigonometrically concepts. Trigonometric Ratios The ratio of the sides of a right-angled triangle with respect to its angles are called trigonometric ratios. Left us draw a right angle triangle say triangle ABC show in figure. Page 3

4 Page 4

5 Page 5

6 Page 6

7 Page 7

8 Trigonometric Ratios of Certain Angles We have defined sin, cos, tan etc. for any acute angle Q but we have not yet found value of sin, cos, tan etc, for even one specific angle. We can use knowledge of geometry to find the values of the trigonometric ratios of some angle say 35, 45, 60, 90 and 0. Trigonometric Ratio of 30 Consider an equilateral triangle ABC. Since each angle in an equilateral triangle, therefore, A, B, C = 60o and AB = BC = CA. Page 8

9 Page 9

10 Page 10

11 Page 11

12 Page 12

13 Page 13

14 Page 14

15 Page 15

16 Height and Distance = -1 = 1 tanθ-secθ tanθ-secθ which is the RHS of the identity, we are required to prove. Let us consider of previous chapter, which is redrawn below. 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. Now, consider the situation. 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. Example: A tower stands vertically on the ground. From a point of the ground, which is 15m away from the foot of the tower, the angle of elevation of the top of the tower is found to be 60O. Find the height of the tower. Page 16

Solution: First let us draw a simple diagram to represent the problem (see Fig.). Here AB represents the tower, CB is the distance of the point from the tower and ACB is the angle of elevation.

17 Solution: First let us draw a simple diagram to represent the problem (see Fig.). 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. Example: 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. 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? Solution: In Fig. the electrician is required to reach the point B on the pole AD. Page 17

18 Page 18

19 Page 19

20 Page 20

21 Page 21

22 Page 22

23 Page 23

24 Page 24

Trigonometry Applications

Name: Date: Period Trigonometry Applications Draw a picture (if one is not provided), write an equation, and solve each problem. Round answers to the nearest hundredths. 1. A 110-ft crane set at an angle