130-Basics of Trigonometry - 16 Important key points

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Basics of trigonometry

Now we will study some examples of trigonometry. 

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ANIL SATPUTE

129-Basics of Trigonometry - 15 Important key points

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Basics of trigonometry

Now we will study some examples of trigonometry. 

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ANIL SATPUTE

128-Basics of Trigonometry - 14 Important key points

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Basics of trigonometry

Now we will study some examples of trigonometry. 

111-Basics of Trigonometry - 13 Important key points

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Basics of trigonometry

Now we will study some examples of trigonometry. 

Example-7

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ANIL SATPUTE

110-Basics of Trigonometry - 12 Important key points

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Basics of trigonometry

Now we will study some examples of trigonometry. 

Example-5

Example-6

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109-Basics of Trigonometry - 11 Important key points

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Basics of trigonometry

Now we will study some examples of trigonometry. 

Example-3

Example-4

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108-Basics of Trigonometry - 10 Important key points

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Basics of trigonometry

Now we will study some examples of trigonometry. 

Example-1

Example-2

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ANIL SATPUTE

102-Tool to calculate difference between two dates (Age calculator)

Click on the following link:

A tool to get the difference between two dates

We are about to explore a handy tool that helps determine the gap between two dates, expressed in years, months, and days. To get started, let’s assess your current age — it’s precise and interactive!
👉 Click on the image below to begin:

Simply enter the necessary details in the Start Date and End Date fields, ensuring the start date precedes the end date. This tool calculates the duration between two dates in years, months, and days.

Whether you're interested in your current age, the age you were when you graduated, or the age gap between you and a relative, this calculator delivers quick and precise results. It's a straightforward yet effective tool for various practical scenarios.

 ANIL SATPUTE

93-Software to calculate Degree and Radians

a) Click here for "01-Degree-2-Degree-Minute-Second-n-Radian-converter"

b) Click here for "02-Degree-Minute-Second-2-Degree-n-Radian-converter"

c) Click here for "03-Radian-2-Degree-n-Degree-Minute-Second-converter"

1) Directed Angles:

The initial arm rotates through a certain amount of rotation in the clockwise or anti-clockwise direction to the terminal arm then this amount of rotation is called the measure of the directed angle and such an angle is called the directed angle.

The directed angle AOB has ray OA as an initial arm and ray OB as the terminal arm. O is called a vertex of an angle AOB. 

Note: 

1) Here, Angle AOB ≠ Angle BOA even if they have the same amount of rotation.

2) If the rotation of the initial arm is anticlockwise, the directed angle is positive and if it is clockwise then the angle is negative.

See the figure carefully to understand the concept.

86-Area Volume Calculation Tool

Special Note: The Following link will help you to get all the calculations that are based on two values given by you.

The most amazing part of this blog is that you will get all calculated values such as perimeters, area volume, curved surface area, and total surface area, using the software of which the link is provided here.

Click on the following Button for Area Volume Calculator:

47-Solution on perpendicular Bisector: Important key points

Today we will see the answer to the question asked in Blog-19.

The question was:

Draw the perpendicular bisector to a line segment drawn at the bottom of the page as shown in the following diagram.

We know the procedure to draw the perpendicular bisector of a line segment.

1) In a compass,  take a distance of more than half the segment. [Here my question is why do we need to take the distance more than half? ]

Here the Answer is very simple. You will come to know it very soon while discussing the basics of drawing perpendicular bisector on the line segment.

2) Taking Point A as the center and radius as more than half the distance of the segment, Draw the arcs on both sides of the segment.

3) Repeat the same by taking point B as the center and with the same radius.

Now you will get the answer to the question of why we need to take the radius as more than half. The arcs drawn with center A and center B should intersect each other.

4) Then join these two points to draw the line which is the perpendicular bisector of the segment AB.

As per Geometry is concerned, we can draw the perpendicular bisector in the same way.

Here we must understand the Basic Concept of a perpendicular bisector. It is already available in the procedure itself. As we are drawing the arcs with the same radius and the centers as point A and point B on both sides of the segment to get two points which are the point of intersection of these arcs. 

As per the question, so many students say their views are as follows.

Some students said that the question is wrong as the line segment on which the perpendicular bisector is to be drawn is at the bottom of the paper. It must be in the middle of the paper.

Some students said that the perpendicular bisector is not possible as we can't draw the intersecting arcs below the segment. The segment should be a little bit above.

Some students said that some additional blank paper needs to place below the line segment and construct the perpendicular bisector.

I was surprised by listening to these answers. The above problem can be solved simply by applying the basics of the perpendicular bisector.

We must know that every point of the perpendicular bisector is equidistant from the endpoints of the segment. So do the following steps to draw the perpendicular.

1) Take a radius of more than half of the line segment AB.

2) Take point A as the center, and draw an arc above segment AB.

3) Take point B as the center, and draw an arc above segment AB.

4) Name the point of intersection of these arcs as point C.

5) Increase the radius and draw two intersecting arcs with centers A and B above the Point C

6) Name this point of intersection of these arcs as point D.

7) Draw line joining points C and D.

Please see the following diagram to understand more about the concept.

Here Points C and D are at equidistant from the endpoints A and B. So line CD is the perpendicular Bisector of line segment AB.

We all together think properly to understand the basic concepts of Math. We will definitely change the entire world to understand better the basics of Math along with other subjects.

Anil Satpute

41-Easy Method to check your calculations

Today we will see an easy method to check your calculations.

In Number system, every number is formed by digits. For our method, we require to get the digit addition (that is Addition of all the digits in the number). 

Now we will see some example of digit addition.

Example-01:  Let the given Number be 795832. The digit addition of this number means to get an addition of all the digit till single digit. 
      
           Digit Addition = 7 + 9 + 5 + 8 + 3 + 2
                                    = 34
                                    = 3 + 4
                                    = 7
 So here Digit addition of the Number 795832 is 7 (which is in the lowest form)

Now we will see still an easy method to do this digit addition. See the following steps to do the fast digit addition.

Step-01: Take out all 9's from the given number if any. In the given number 795832, one 9 is observed, so take it away from the number, So now we have the number for digit addition will be 75832. (Note: The removal of all 9's is only for getting digit addition. For original calculation we must take the complete number).

Step-02: Take out all the digits whose addition comes out to be 9. in our resulted number in step-01, 75832, if we observe very carefully, we will notice that the digits 7 and 2 gives addition as 9 so remove these two digits, so now we have the number for digit addition will be 583.

Step-03: Now we need to take the addition of all the remaining digits. Here we have 
           Digit Addition = (5 + 8) + 3
                                    = 16
                                    = 1 + 6
                                    =  7
Which is as the addition of all the digits calculated by adding all the digits and then reducing it till single digit.

Now we will see all the above 3 steps for getting addition.

           Digit Addition = 7 + 9 + 5 + 8 + 3 + 2
                                    = (5 + 8) + 3
                                    = 13 + 3
                                    = 16
                                    = 1 + 6
                                    = 7
Example-02: Find the digit addition of the Number 543249815
           Digit Addition = 5 + 4 + 3 + 2 + 4 + 9 + 8 + 1 + 5
                                    = (5 + 4) + (3 + 2 + 4) + (9) + (8 + 1) + 5    
                                        (All terms in the brackets gives addition as 9)
                                    =  5

So the digit addition of 543249817 is 5.

Now we will see the procedure for checking our calculations.

Example-01: Add the three numbers 9432, 7859 and 8147

                         9432
                         7859
                     +  8147
                 -----------------
                       25438

Now we check our calculations by using digit addition method.

A) Digit addition of the Numbers = (9 + 4 + 3 + 2) + (7 + 8 + 5 + 9) + (8 + 1 + 4 + 7)
                                                         = (4 + 3 + 2) + (7 + 8 + 5) + (4 + 7)
                                                         = (9) + (20) + (11)
                                                         = (2 + 0) + (1 + 1)
                                                         = (2) + (2)
                                                         =  4
B) Digit addition of our result      = (2 + 5 + 4 + 3 + 8)
                                                         = (2 + 3 + 8)
                                                         = (5 + 8)
                                                         = (13)
                                                         = (1 + 3)
                                                         =  4
So our answer of addition is correct.
Example-02: Find the Multiplication of 279 and 563 
(Please Click on  "Simple Method of Multiplication" to learn a new method of multiplication, We will take the same example here to check the result)

A) Digit addition of the Numbers 
            = (2 + 7 + 9) x (5 +  6 + 3) 
            = (9) x (5)
            = (45)
            = (4 + 5)
            = (9)
B) Digit addition of our result
            = (1 + 5 + 7 + 0 + 7 + 7)
            = (6 + 21)
            = (6 + 3)
            = (9)

So we can say our multiplication is right.

For Subtraction also we need to transfer the negative term to another side only to check our result. 

For Example, If we would like to check the calculations for the subtraction A - B = C, then for checking purpose we need to take this subtraction as 
A = B + C .

The same thing is to be done for the division. (In the division, we need to transfer the division into the product form)

For Example, If we got the result as X / Y = Z, then we need to check our calculations for 
X = YZ.

Please study the above method & try your self by doing some more operations with the help of your parents. You can also ask me your query if any by filling the Entry form available at the top left the tab of this blog.

Enjoy Learning Mathematics by creating such type of interesting things.

Anil Satpute

37-Easy Method For Creating Mathematical Table

Today I will publish my blog on the article of my younger son Master Abhishek Anil Satpute. Now he is studying in the 11th Grade of Science Stream. In 2008, when he was in 7th Grade (Standard), he found one interesting method: Mathematical Tables.  My elder son Master Aaswad Anil Satpute also found out so many interesting things in Mathematics since his 4th Grade (Standard). (Now he is in the last semester of Computer Engineering)

This Article by Abhishek Anil Satpute is written in his own way. I thank him for allowing me to publish his article.

Easy method for creating mathematical tables

Abhishek Anil Satpute

7th Grade (2008)

While studying the mathematical tables, I found one interesting method for creating 
the tables.

  A)  New Definitions:

  1) Small Friends: Two numbers having their addition as 5 are 
      known as Small Friends

  2) Normal Friends: Two numbers having their addition as 10 are
       known as Normal Friends.

  3) Big Friends: Two numbers having their addition as 15 are 
      known as Big Friends.

B) Table of friends:

	Small Friends	Normal Friends	Big Friends
1	4	9	14 NA
2	3	8	13 NA
3	2	7	12 NA
4	1	6	11 NA
5	0	5	10 NA
6	-	4	9
7	-	3	8
8	-	2	7
9	-	1	6

c) Procedure for creating unit-placed digits:

1)  Take the number of which the table is to create.                           (A)

2)  Write the unit digit of the double of the unit placed digit of (A).      (B)

3)  Write the friend number of the unit digit of (B) as given below.      (C)

Digit in the units place	Type of friends to be used
1, 5 or 7	Small Friends (05)
2, 4, 6 or 8	Normal Friends (10)
3 or 9	Big Friends (15)

4)   Write the friend number of the unit digit of (A) as given below. (D)

Digit in the units place	Type of friends to be used
1, 3, or 5	Small Friends (05)
2, 4, 6 or 8	Normal Friends (10)
7 or 9	Big Friends (15)

5) Write 5 if (A) is odd & 0 if (A) is even.

6) Write the normal friend number of the unit digit of (D).

7) Write the normal friend number of the unit digit of (C).

8) Write the normal friend number of the unit digit of (B).

9) Write the normal friend number of the unit digit of (A).

10) Write the 10th multiple of (A).

D)     Procedure for creating ten’s/hundred's placed digits:

Now we will write the ten’s placed digits of the table. We will follow a few simple 

Rules as given below.

1)  If ten’s placed digit of (A) is “0”, then while writing the table for (A), for Increasing 

the digit in unit place, increment the ten’s placed digit by “0” else by “1”

2) If ten’s placed digit of (A) is “1”, then while writing the table for (A), for Increasing 

a digit in unit place, increment the ten’s placed digit by “1” else by “2”

3) If ten’s placed digit of (A) is “2”, then while writing the table for (A), for Increasing 

a digit in unit place, increment the ten’s placed digit by “2” else by “3” and so on.

E)     Examples: 

1)  Table of 06 (A) Unit place is 6, so for the 3rd step take Normal Friends, for the 4th step take Normal Friends, and the ten’s placed digit is 0, so increment either by 0 or by 1.

01	Take the number 06 (A)	06
02	Double is 12, so take 2 (B). 2<6 so 0+1=1, so write 12.	12
03	Normal-friend of 2 is 8 (C) so take 8. 8>2 so 1+0=1, so write 18	18
04	Normal-friend of 6 is 4 (D), so take 4. 4<8 so 1+1=2, so write 24	24
05	6 (A) is Even so take 0, 0<4 so 2+1=3 so write 30	30
06	Normal-friend of (D) 4 is 6, so take 6, 6>0 so 3+0=3, so write 36	36
07	Normal-friend of (C) 8 is 2, so take 2, 2<6 so 3+1=4, so write 42	42
08	Normal-friend of (B) 2 is 8, so take 8, 8>6 so 4+0=4, so write 48	48
09	Normal-friend of (A) 6 is 4, so take 4, 4<6 so 4+1=5, so write 54	54
10	Always take 0, 0 < 4, so 5+1=6, so write 60	60

2)      Table of 27 (A): Unit's place is 7, so for the 3rd step take Small Friends, for 

     4th step takes Big Friends, and the ten’s placed digit is 2, so increment either 

       by 2 or by 3.

01	Take the number 27 (A)	27
02	Double is 14, so take 4 (B). 4<7 so 2+3=5, so write 54.	54
03	Small-friend of 4 is 1 (C) so take 1. 1<4 so 5+3=8, so write 81	81
04	Big friend of 7 is 8 (D), so take 8. 8>1 so 8+2=10, so write 108	108
05	7 (A) is Odd so take 5, 5<8 so 10+3=13 so write 135	135
06	Normal friend of 8 (D) is 2, so take 2, 2<5 so 13+3=16, so write 162	162
07	Normal friend of 1 (C) is 9, so take 9, 9>2 so 16+2=18, so write 189	189
08	Normal-friend of 4 (B) is 6, so take 6, 6<9 so 18+3=21, so write 216	216
09	Normal-friend of 7 (A) is 3, so take 3, 3<6 so 21+3=24, so write 243	243
10	Always take 0, 0<3, so 24+3=27, so write 270	270

Once again I thank Master Abhishek for his article.

Anil Satpute