# Standard Symbols

Following symbols in relation to Î”

*ABC*are universally adopted.*m*âˆ

*BAC*=

*A*,

*m*âˆ

*ABC*=

*B*,

*m*âˆ

*BCA*=

*C*

*A*+

*B*+

*C*=

*Ï€*

*AB*=

*c*,

*BC*=

*a*,

*CA*=

*b*

- Semi-perimeter of the triangle =
*s*= So*a*+*b*+*c*= 2*s.* - The radius of the circumcircle of the triangle, that is, circumradius =
*R*. - The radius of the incircle of the triangle, that is, inradius =
*r*. - Area of the triangle =
*S =*Î”.

# Sine rule

The sine rule is .

**Nepierâ€™s formula**

- tan
- tan
- tan

# Cosine rule

In a Î”

*ABC*, we have cos*A*= ,cos

*B*= , cos*C*= .

*Notes:*- The above proof will not change even if âˆ
*A*is a right angle or an obtuse angle. - If the lengths of the three sides of a triangle are known, we can find all the angles by using cosine rule because this rule gives us cos
*A*, cos*B*, and cos*C*. We know that*A*,*B*,*C*are in (0,*Ï€*) and the cosine function is one-one in [0,*Ï€*]. So*A*,*B*,*C*are precisely determined. Similarly, if two sides (say*b*and*c*) and the included angle*A*are given, the cosine rule cos*A*= will give us*a*and then knowing*a*,*b*,*c*we can find*B*and*C*by the cosine rule.

**Projection formula**

*a*=

*c*cos

*B*+

*b*cos

*C*

*b*=

*a*cos

*C*+

*c*cos

*A*

*c*=

*a*cos

*B*+

*b*cos

*A*

**Half angle formulas**

- i. sin
- i. cos