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Standard Symbols

Following symbols in relation to ΔABC are universally adopted.
mBAC = A, mABC = B, mBCA = C
A + B + C = π
AB = c, BC = a, CA = b
  • Semi-perimeter of the triangle = s = 73076.png So a + b + c = 2s.
  • 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 = Δ.

Sine rule

The sine rule is 72718.png.
Nepier’s formula
  1. tan72712.png
  2. tan72706.png
  3. tan72700.png

Cosine rule

In a ΔABC, we have cos A = 72694.png,
cos B = 72688.png, cos C = 72682.png.


  • The above proof will not change even if ∠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 ABC are in (0, π) and the cosine function is one-one in [0, π]. So ABC are precisely determined. Similarly, if two sides (say b and c) and the included angle A are given, the cosine rule cos A = 73874.png will give us aand then knowing abc 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
  1. i. sin 72670.png
    ii. sin 72662.png
    iii. sin 72656.png
  2. i. cos 72650.png
    ii. cos 72644.png
    iii. cos 72637.png

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