Concrete Introduction to Trigonometry
Are you sitting in a room? Just look across. Do you see a wall? Let's say this can be represented like the image to the side. If I ask you to measure the height, how can you do it? Which is easier to measure? The length or the height? It's the
Obviously it's the length because it is on the floor. You can use a tape or just measure with your feet.
But how can you climb up a wall to measure the height. We need better tools and methods to measure heights of tall things like mountains or buildings.
Now, please draw a line from "Person" point to "Wall Bottom" point.
Next draw a line from "Person" point to "Wall Top" point.
What do you see? It's in the shape of aVery good. As you can see triangles are every where Now let's see what else we can deduce. We can see that length of the floor is 6 cm and when the person look up at the ceiling his eyes make a 45o angle. We also know that the angle between the wall and the floor is
So, we have a right angle triangle and one of it's angle is 45o. That makes this an
And from the properties of right angle isosceles triangles we know that opposite side = adjacent size. In our scenario we see that the adjacent side(floor) length is 6 cm. So the opposite side(height) should be
Congratulations! By observing that we have a triangle shape in the real world and using our knowledge of triangles we were able to calculate the height of the wall without climbing it. Let us now move onto a formal introduction to Trigonometry.
Let us take some examples from our surroundings where right triangles can be imagined to be formed. For instance :_
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. Can the student find out the height of the Minar, without actually measuring it?
Figure for Qutub Minar
- 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. If you know the height at which the person is sitting, can you find the width of the river?
Person sitting on a river bank
- 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 intially it was at point A. When both the mother and daughter came out to see it, it had already travelled to another point B. Can you find the altitude of B from the ground?
Viewing a hot air balloon in the sky.
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 trigonometrical concepts.
In this chapter, we will study some ratios of the sides of a right triangle with respect to its acute angles, called trigonometric ratios of the angle. We will restrict our discussion to acute angles only. However, these ratios can be extended to other angles also. We will also define the trigonometric ratios for angles of measure 0° and 90°. We will calculate trigonometric ratios for some specific angles and establish some identities involving these ratios, called trigonometric identities.