74-Basics of Trigonometry - 02 Important key points

Click here for the previous basics of trigonometry.

Trigonometry is an important topic for your higher studies. We must study this topic with all basics. Actually, this topic is very simple. If we concentrate to learn this topic from the beginning, we will definitely understand the basics of this topic and it will help us to remember the formulas (formulae) very easily. At the end of all the Blogs related to “Basics of trigonometry”, we will list all the formulas (formulae) in a separate sheet so that it will be helpful for you to remember all these formulas (formulae).

1) Define all trigonometric ratios for an angle C of the above triangle ABC.  

Answer: 
1) sin C  = AB/AC
2) cos C = BC/AC
3) tan C  = AB/BC
4) csc C = AC/AB
5) sec C  = AC/BC
6) cot C  = BC/AB  

2) Observe the adjacent figure and write down all the trigonometric ratios.
Answer:
1) sin P   = RQ/PQ          1) sin Q   = PR/PQ
2) cos P  = PR/PQ           2) cos Q  = RQ/PQ 
3) tan P   = RQ/PR           3) tan Q  = PR/RQ
4) csc P  = PQ/RQ           4) csc Q  = PQ/PR
5) sec P  = PQ/PR            5) sec Q = PQ/RQ
6) cot P   = PR/RQ           6) cot Q  = RQ/PR  
In the above example, just observe 
case 1) sin P and cos Q
case 2) cos P and sin Q

Here, the basic point to understand is that the comparison of two angles is given with different trigonometric ratios. First, observe these two angles. angle P and angle Q. In a right-angled triangle, QPR, Angle R = 90, so the remaining two angles P and Q are complementary angles. 

Here all of us will remember the following trigonometric relations of complementary angles of a right-angled triangle.

1) sin A  = cos (90-A)
2) cos A = sin (90-A)
3) sec A = csc (90-A)
4) csc A = sec (90-A)
5) tan A  = cot (90-A)
6) cot A  = tan (90-A)

A) An easy way to remember the above relations:

The relation between the trigonometric ratios of co-terminal angles. (See the following figure)

Trigonometric Ratios of an angle A	Trigonometric Ratios of Complimentary Angle of A. (90°-A)	Ratio (In the form of the length of the sides of a triangle)
sin A	cos (90°-A)	BC/AC
cos A	sin (90°-A)	AB/AC
csc A	sec (90°-A)	AC/BC
sec A	csc (90°-A)	AC/AB
tan A	cot (90°-A)	BC/AB
cot A	tan (90°-A)	AB/BC

B) Relation between the trigonometric ratios:

Observe the following trigonometric ratios from the figure given below:

1) sin C  = AB/AC

2) cos C = BC/AC

3) tan C  = AB/BC

4) csc C = AC/AB

5) sec C  = AC/BC

6) cot C  = BC/AB

If we divide equation 1 by equation 2 we get equation 3. i.e. 

                            

In the same way, we can prove that "cos C / sin C  =  cot C"

In the Next Blog, we will study the trigonometric ratios of some standard angles and trigonometric identities.

Click here for the next basics of trigonometry.

Anil Satpute

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.

Click here for the next basics of trigonometry.

Anil Satpute

72-Magic Square-5 (different cells and addition part-2)

Click here for the next blog on the magic square.

Addition of different cell elements of Magic Squares:

🔢 Delving Deeper into Hidden Patterns in the Magic Square  

We have already observed that in a 4 × 4 magic square, the total numbers in every row, column, and diagonal consistently add up to 34.  

That’s the charm of a magic square!  

Each of these lines comprises four elements; this unchanging total is one of its key characteristics.  

However, what’s even more intriguing is that the sum of numbers arranged in particular patterns or positions throughout the square, in addition to rows, columns, or diagonals, also totals 34.  

📐 The illustration below highlights various combinations of 4 cells that adhere to this unique property.  

It’s akin to uncovering concealed symmetry and mathematical harmony that goes beyond the obvious.

 (1 + 15 + 12 + 6)      (14 + 4 + 7 + 9)

(R₁C₁, R₁C₂, R₂C_1, R₂C₂)    (R₁C₃, R₁C₄, R₂C_3, R₂C₄)

🔍 Discovering More Concealed Groups in the Magic Square

In the provided example, the total of the chosen numbers — 1 + 15 + 12 + 6 — equals 34.

This verifies the magical characteristic even when numbers are selected from non-linear positions!

Similarly, numerous other combinations of 4 cells, located in various square areas, also yield the same sum.

✨ Now let's delve into additional groups where the numbers may not align in a straight line, yet their sum still fantastically amounts to 34.

These instances beautifully showcase the hidden symmetry and refined structure of a magic square.

 

(11 + 5 + 2 + 16)      (8 + 10 + 13 + 3)

(R₃C₃, R₃C₄, R₄C_3, R₄C₄)  (R₃C₁, R₃C₂, R₄C_1, R₄C₂)

(15 + 14 + 3 + 2)      (8 + 12 + 9 + 5)

(R₁C₂, R₁C₃, R₄C_2, R₄C₃)  (R₂C₁, R₃C₁, R₂C_4, R₃C₄)

(15 + 12 + 5 + 2)      (14 + 9 + 8 + 3)

(R₁C₂, R₂C₁, R₃C_4, R₄C₃)  (R₁C₃, R₂C₄, R₃C_1, R₄C₂)

(6 + 7 + 10 + 11)         (1 + 4 + 13 + 16)

(R₂C₂, R₂C₃, R₃C_2, R₃C₃)  (R₁C₁, R₁C₄, R₄C_1, R₄C₄)

(1 + 7 + 10 + 16)         (4 + 6 + 11 + 13)

(R₁C₁, R₂C₃, R₃C_2, R₄C₄)  (R₁C₄, R₂C₂, R₃C_3, R₄C₁)

(6 + 9 + 3 + 16)         (15 + 4 + 10 + 5)

(R₂C₂, R₂C₄, R₄C_2, R₄C₄)  (R₁C₂, R₁C₄, R₃C_2, R₃C₃)

(12 + 7 + 13 + 2)         (1 + 14 + 8 + 11)

(R₂C₁, R₂C₃, R₄C_1, R₄C₃)  (R₁C₁, R₁C₃, R₃C_1, R₃C₃)
✨ Explore More Magical Patterns!
In the same way, you can discover many intriguing relationships between the numbers placed in different cells and their resulting sums. Keep exploring — every pattern reveals a new layer of mathematical beauty!

Click here for the next blog on the magic square.

ANIL SATPUTE