Trigonometry/Solving triangles by half-angle formulae
In this section, we present alternative ways of solving triangles by using half-angle formulae.
Given a triangle with sides a, b and c, define
- s = 1⁄2(a+b+c).
- a+b-c = 2s-2c = 2(s-c)
and similarly for a and b.
We have from the cosine theorem
By symmetry, there are similar expressions involving the angles B and C.
Note that in this expression and all the others for half angles, the positive square root is always taken. This is because a half-angle of a triangle must always be less than a right angle.
Cos(A/2) and tan(A/2)
Again, by symmetry there are similar expressions involving the angles B and C.
Sin(A) and Heron's formula
A formula for sin(A) can be found using either of the following identities:
These both lead to
The positive square root is always used, since A cannot exceed 180º. Again, by symmetry there are similar expressions involving the angles B and C. These expressions provide an alternative proof of the sine theorem.
Since the area of a triangle
which is Heron's formula.