The three sides a, b, c and the three angles A, B, C are called the elements of the triangle ABC. When any three of these six elements (except all the three angles) of a triangle are given, the triangle is known completely; that is, the other three elements can be expressed in terms of the given elements and can be evaluated. This process is called the solution of triangles.
- If the three sides a, b, and c are given, angle A is obtained from
or
. B and C can be obtained in a similar way.
- If two sides b and c and the included angle A are given, then
gives
. Also,
, so that B and C can be evaluated. The third side is given by
, or,
.
- If two sides b and c and the angle B (opposite to side b) are given, then
. And,
, and
give the remaining elements.
By applying the cosine rule, we have:
, or if we manipulate this, we get

or, 
This equation leads to the following cases:
Case I:
If
, no such triangle is possible.
Case II:
Let
. There are further following two cases:
Sub-case II a:
B is an obtuse angle, that is,
is negative. There exists no such triangle.
Sub-case II b:
B is an acute angle, that is,
is positive. There exists only one such triangle.
Case III:
Let
. There are following two cases further here also:
Sub-case IIIa:
B is an acute angle, that is,
is positive. In this case, two values of a will exist if and only if
or,
, which means two such triangles are possible. If
, only one such triangle is possible.
Sub-case IIIb:
B is an obtuse angle, that is,
is negative. In this case, triangle will exist if and only if
. So, in this case, only one such triangle is possible. If
, there exists no such triangle.
Note:
If one side a and angles B and C are given, then
, and
and
.
If the three angles A, B and C are given, we can only find the ratios of the three sides a, b, and c by using the sine rule(since there are infinite number of similar triangles possible).
More theory later,
Nalin Pithwa