There are several options of finding of sizes of all corners in a triangle if lengths of three of its parties are known. One of ways consists in use of two different formulas of calculation of the area of a triangle. It is also possible to apply to simplification of calculations the theorem of sine and the theorem of the sum of corners of a triangle.
1. Use, for example, two formulas of calculation of the area of a triangle only three of its known parties (Heron's formula) are involved in one of which, and in another - two parties and a sine of the angle between them. Using different couples of parties in the second formula, you will be able to determine sizes of each of triangle corners.
2. Solve a problem in a general view. Heron's formula determines the area of a triangle as a square root from the work of a poluperimetr (a half from the sum of all parties) on differences between poluperimetry and each of the parties. If to replace perimeter with the sum of the parties, then the formula can be written down in such look: S=0.25 ∗√ (a+b+c) ∗ (b+c-a) ∗ (a+c-b) ∗ (a+b-c). From other party the area of a triangle can be expressed as a half of the work of two of its parties on a sine of the angle between them. For example, for the parties of an and b with a corner γ between them this formula can be written down so: S=a∗b∗sin(γ). Replace the left part of equality with Heron's formula: 0.25 ∗√ (a+b+c) ∗ (b+c-a) ∗ (a+c-b) ∗ (a+b-c) of =a∗b∗sin(γ). Bring out of this equality a formula for a sine of the angle γ: sin(γ) =0.25 ∗√ (a+b+c) ∗ (b+c-a) ∗ (a+c-b) ∗ (a+b-c) / (a∗b ∗)
3. Similar formulas for two other corners: sin(α) =0.25 ∗√ (a+b+c) ∗ (b+c-a) ∗ (a+c-b) ∗ (a+b-c) / (b∗c ∗) sin(β) =0.25 ∗√ (a+b+c) ∗ (b+c-a) ∗ (a+c-b) ∗ / (a∗c ∗) Instead of these formulas can use (a+b-c) the theorem of sine from which follows that ratios of the parties and sine opposite corners are equal to them in a triangle. That is, having calculated a sine of one of corners in the previous step, it is possible to find a sine of other corner on simpler formula: sin(α)=sin(γ) ∗ a/c. And that the sum of corners is equal in a triangle 180 ° the third corner can be calculated even more simply: β=180 °-α-γ.
4. Use, for example, the standard Windows calculator for finding of sizes of corners in degrees after you calculate values of sine of these corners by formulas. That to make it, apply trigonometrical function, the return to a sine - an arcsine.