Trigonometry/Solving triangles by half-angle formulae

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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 = 12(a+b+c).

Note that

a+b-c = 2s-2c = 2(s-c)

and similarly for a and b.

We have from the cosine theorem

Sin(A/2)Edit

So

.

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)Edit

So

.
.

Again, by symmetry there are similar expressions involving the angles B and C.

Sin(A) and Heron's formulaEdit

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.