Triangle Formula





Triangle Formula

A triangle is a basic geometrical shape, with consists of three corners or vertices and three sides which are called line segments. There are various types of triangles some of them are :- Equilateral triangle, isosceles triangle, Scalene triangle, Right triangle, Special right triangle etc. There are three angles in a triangle and the sum of angles  within a triangle is equal to 180 degree. In simple word’s a triangle can be described as a 3 sided polygon and sometimes is is also called as trigon.

 
Some of the important formulas and Law’s related to triangle are described below:-

 
1.The cosine law

The cosine law is a useful tool for solving the problems of triangles; the cosine law can be applied in any kind of triangle. The cosine law states that “the square of one plane side is equal to the sum of the other two sides minus twice the product of other two sides and the cosine angle between them”.
The cosine law is useful for finding third side if two sides and angle between them are known and for finding the angle of a triangle if three sides are known.

In any triangle ABC

I.\cos A = \dfrac{b^2 + c^2- a^2}{2bc} or  a^2 = b^2 + c^2- 2bc\cos A

II.\cos B = \dfrac{c^2 + a^2- b^2}{2ca} or b^2 = c^2 + a^2- 2ac\cos B

III.\cos C = \dfrac{a^2 + b^2-c^2}{2ab} or c^2 = a^2 + b^2- 2ab\cos C

 

2.The sine law

 

In trigonometry, the sine law is an equation relating length of the sides of the triangle and the angle between them. The sine law is commonly applied for finding the sides and the angle between them. The sine law gives the following equation,

\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}

 

3.The projection law

 

In any triangle ABC

b \cos C + c \cos B = a \\ c \cos A + a \cos C = b \\ a \cos B + b \cos A = c

 

4.Tangent law

 

In any triangle ABC

 

\tan \dfrac{1}{2} ( B- C) = \dfrac{b- c}{b + c} \cot \dfrac{1}{2} A

 

\tan \dfrac{1}{2}(C- A) = \dfrac{c- a}{c + a} \cot \dfrac{1}{2} B

 

\tan \dfrac{1}{2} (A- B) = \dfrac{a- b}{a + b} \cot \dfrac{1}{2} C

 

5.The half angle formulas

 

In any triangle ABC

\sin \frac{A}{2} = \sqrt{\dfrac{(s- b)(s- c)}{bc}} ; \, \, \, \cos \frac{A}{2} = \sqrt{\dfrac{s(s- a)}{bc}} \\ \sin \frac{B}{2} = \sqrt{\dfrac{(s- c)(s- a)}{ca} } ; \, \, \, \cos \frac{B}{2} = \sqrt{\dfrac{s(s- b)}{ca}}\\ \sin \frac{C}{2} = \sqrt{\dfrac{(s- c)(s- b)}{ab}} ; \, \, \, \cos \frac{C}{2} = \sqrt{\dfrac{s(s- c)}{ab}}\\ \tan \frac{A}{2} = \sqrt{\dfrac{s- b) (s- c)}{s(s- a)}} ; \, \, \, \tan \frac{B}{2} = \sqrt{\dfrac{(s- c)(s- a)}{s(s- b)}} \\ \tan \frac{C}{2} = \sqrt{\dfrac{(s- a)(s- b)}{s(s- c)}}
where s = \dfrac{a + b + c}{2}

 

6.The area of triangle

i. \Delta = \sqrt{s(s- a)(s- b)(s- c)}\\[3mm] ii. \Delta = \dfrac{1}{2}bc \sin A = \dfrac{1}{2}ca \sin B = \dfrac{1}{2}ab \sin C \\[3mm] iii. \Delta = \dfrac{abc}{4R}\, \\[3mm] iv. \sin A = \dfrac{2 \Delta}{bc} \\ v. \tan \dfrac{1}{2} A = \dfrac{(s- b)(s- c)}{\Delta} \, \\[3mm] vi. \cot \dfrac{1}{2} A = \dfrac{s(s- a)}{\Delta}
Where R = \dfrac{a}{2 \sin A}

 

7.The in-circle (inscribed circle)

 

i. r = \dfrac{\Delta}{s} \\[3mm] ii. r = a \sin \frac{B}{2} \sin \frac{C}{2} \sin \frac{A}{2} \\[3mm] iii. R = 4R \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} \\[3mm] iv. R = (s- a) \tan \frac{A}{2} \\[3mm] v. \tan \dfrac{1}{2} A = \dfrac{(s- b)(s- c)}{\Delta} etc.

 

8.The ex- circle (escribed circle)

 

i. r_1 = \dfrac{\Delta}{(s- a)} = 4R \sin \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2} \\ ii. r_2 = \dfrac{\Delta}{(s- b)} = 4R \cos \frac{A}{2} \sin \frac{B}{2} \cos \frac{C}{2} \\ iii. r_3 = \dfrac{\Delta}{(s- c)} = 4R \cos \frac{A}{2} \cos \frac{B}{2} \sin \frac{C}{2}

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