Proof of Law of Sine

The Law of Sine

The Law of Sine states that given any triangle with side lengths a, b and c and opposing angles A, B and C

First we'll divide the triangle into two right triangles by drawing the line that passes through A and is perpendicular to a 

From the definition of sine ;

or 

Since both are equal to h ; we can say that :

Now dividing by sinB and sinC , gives us ;

Hence Proved ! 

Now you give it a try it for other sides and ratios to further prove !

                                             

Proof of Cosine Rule

The Law of Cosines

The Law of Cosines states that given any triangle with side lengths a, b and c and opposing angles A, B and C is such that :

First we divide the triangle into two right triangles by drawing the line that passes through B and is perpendicular to b.

         

Now we have a right triangle with c as the hypotenuse. And so according to the Pythagorean Theorem

  

Is 1 a prime number?

Is 1 a Prime Number? 

Explained using The Fundamental Theorem of Arithmetic

Every time you come across a discussion of prime numbers, or a question requiring use of prime numbers, you might get stuck at this question in your head : Is 1 a Prime Number? 

 The definition of a prime number is : a number that is only divisible by one, and itself.
Considering the definition, it seems that 1 is a prime number. After all, it is divisible by one, and it is divisible by itself (that is 1), and hence it was thought that 1 is a prime number. However, mathematicians had to exclude 1 from the prime numbers due to very important theorem that exists in mathematics. 

Before answering the question "Is 1 a prime number" , we need to understand that theorem :

"The Fundamental Theorem of Arithmetic"

The "Fundamental Theorem of Arithmetic" ,discovered by Carl Friedrich Gauss in 1801, states that : "Every integer greater than 1 is either a prime number itself or can be written as a unique product of prime numbers (ignoring the order)." 

The statements makes three important points, discussed below: 

• "Every integer greater than 1 is either a prime number itself .... " : This refers to numbers 2,3,5,7,11,13... i.e all the prime numbers.
   
• "Product of prime numbers.." : This refers to all other numbers (non-prime/composite numbers) e.g 4,6,8,9,10,12,14,15,16... and implies that these numbers can be generated by multiplying some prime numbers.
  
  For example, lets take 15 :
  The prime factorization of 15 is 3 x 5 (Both are prime)
  
  or lets take 10:
  This can be generated by multiplying : 2 x 5 (Both are prime)
  
   Hence all non-prime/composite numbers can be generated by multiplying some prime numbers. Every number has some prime numbers behind it!
  
• "Unique product of prime numbers" : Now this is a very important part of this theorem (which will answer our main question regarding 1) . This means that every non-prime number can be generated by multiplying some prime numbers, but that set of prime numbers is unique for every number. No same number can be generated by multiplying different prime numbers:
  
  Lets take 42 :
  It can be generated by : 2 x 3 x 7 (prime numbers)
  There is no other combination of prime numbers that, when multiplied, result in 42.
  So 2 x 3 x 7 is a unique combination of prime numbers that generates 42.
  
  Lets look at another example:
  18 can be expressed as 2 x 3 x 3 ( prime numbers) or 2 x 3²
  This again is a unique combination of prime numbers, and no other combination is possible.

Now, coming back to our main question : Is 1 a Prime Number? The Answer is NO!
Proof?

This is because the "Fundamental Theorem of Arithmetic" states that for every non-prime number, there is a unique combination of prime numbers that generate that number. Hence there cannot be more than one combination of prime numbers that, if multiplied, result in the same number. 

But if 1 is taken as prime then this does not hold true.
For example, if we take 15, it can be expressed as: 

3 x 5

or 1 x 3 x 5

or 1 x 1 x 3 x 5

or 1 x 1 x 1 x 3 x5

... And there would be infinite number of combinations of prime numbers that generate 15. 

Hence, if 1 were a prime number, then the theorem would be false and the uniqueness would fail.