73-Basics of Trigonometry - 01 Important key points

When an artist wants to draw a picture then he/she always draws a replica of that object. Say, a replica of the great Taj Mahal of Agra (India) or the Statue of Liberty and so on. He/She fixes the drawing paper on the table. Fixes the distance between the paper and his/her eyesight. Then he/she takes all required measurements of the object (Taj-Mahal/Statue of Liberty) on a pencil by holding it between his/her eye-site and the object by locating the bottom of an object at the lower side of the pencil and the tip of an object at the tip of the pencil. Using these distances on the pencil he/she fixes the height, width, and all other dimensions of an object on the drawing paper. Here, using the properties of similarities in a right-angled triangle, an artist or an architecture

can able to draw the exact replica of an object in a smaller size. He/she holds the tip of the pencil adjusting with the tip of the object and fixes the bottom of the object at the lower part of the pencil to fix the height of an object. In the adjacent figure, an artist/architect observes an object BC from point A and marks its height as MN on the pencil. Here ABC is the right-angled triangle with AB as the base, BC as the height, and AC as the hypotenuses. Here the line segments BC and MN are in proportion with AB and AM.
Mathematically, we can say that [BC/AB] = [MN/AM].

Now let us study something about the right-angled triangle. In the adjacent figure triangle, ABC is the right-angled triangle. < ABC is right-angled, so the side opposite to the right angle is known as the hypotenuse. Considering < A is our angle of the right-angled triangle, then side AB is known as the side adjacent to angle A, and side BC is the side opposite to angle A.

Note: There are two types of concepts of trigonometry. We will consider the first concept as a lower-level concept of trigonometry and the other one as a higher-level trigonometry. In lower-level trigonometry, we will study the trigonometric ratios of only the acute angles of the right-angled triangle. Whereas in higher level trigonometry, we will study the trigonometric ratios for any angle. Now let us start with lower-level trigonometry. 

Now, let us start our study of trigonometry. For any right-angled triangle, there are three sides. Let us write down all possible ratios of the lengths of the triangle. 

1) BC/AC

2) AB/AC

3) BC/AB

4) AC/BC

5) AC/AB

6) AB/BC

There are only 6 possible ratios that can be defined from triangle ABC. Considering these 6 ratios, we can develop entire trigonometry. Trigonometric ratios can be defined for a certain angle of the right-angled triangle. Let us understand the words "Side opposite to an angle" or "Side adjacent to an angle". See the above diagram carefully. In the above diagram, we consider angle A as our angle of the right-angled triangle ABC. Side AB is adjacent to angle A and in the same wayside BC is the side opposite to angle A. We will write the above six ratios considering angle A once again using the words "Side opposite to an angle", "Side adjacent to an angle" and "hypotenuse". We will name these ratios in trigonometric forms.

1) BC/AC   side opposite to A/hypotenuse (sin A)

2) AB/AC   side adjacent to A/hypotenuse (cos A)

3) BC/AB   side opposite to A/side adjacent to A      (tan A)

4) AC/BC   hypotenuse /  side opposite to A                 (csc A) 

5) AC/AB   hypotenuse /  side adjacent to A                 (sec A) 

6) AB/BC   side adjacent to A/side opposite to A      (cot A)

Details of trigonometric ratios:
1) sin A  ------  sine of angle A
2) cos A ------  co-sine of angle A
3) tan A  ------  tangent of angle A
4) csc A ------  co-secant of angle A (generally written as cosec A)
5) sec A ------  secant of angle A
6) cot A ------  co-tangent of angle A

Now we will define all six trigonometric ratios as follows:

1) sin A   =  BC/AC  =  opposite / hypotenuse
2) cos A  =  AB/AC  =  adjacent / hypotenuse
3) tan A   =  BC/AB  =  opposite / adjacent
4) csc A  =  AC/BC  =  hypotenuse / opposite
5) sec A  =  AC/AB  =  hypotenuse / adjacent
6) cot A  =  AB/BC  =  adjacent /  opposite

Now study all basic concepts of trigonometry as discussed above and do at least two experiments as stated below.

1) Define all trigonometric ratios for an angle C of the above triangle ABC.
2) Observe the adjacent figure and write down all the trigonometric ratios.

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Anil Satpute