Law of Tangents – Definition – Formula – Proof and Examples

In trigonometry, the law of tangents describes the relationship between the sum and difference of sides of a right triangle and tangents of half of the sum and difference of the angles opposite to the sides.

Formulas for law of tangents

The law of tangents states that 

Where a, b, and c are the lengths of three sides of a triangle and A, B, and C are angle the opposite those three respective sides.

Proof the law of tangents

The law of tangents states that 

We know law of sines 

a/ SinA  =  b/SinB  =  c/SinC

    Let,    a/ SinA  =  b/SinB  =  c/SinC  =  k

∴      a = ksinA,     b = ksinB,      and c = ksinC

 From this   a + b = ksinA + ksinB  and

a – b = ksinA – ksinB
so we get,

Common factor taken out

Simplified using quotient indentities

Similarly for other sides