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

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