Introduction

In many geometric problems, you might be given only partial information about a triangle, such as one side length and the two other angles. This article explores the methods to find the unknown parts of the triangle, including the third angle, when only one side and two angles are known.

Determining the Third Angle

In any triangle, the sum of the interior angles equals 180 degrees (or π radians). Therefore, if you know two of the angles, you can easily find the third. Simply subtract the sum of the known angles from 180 degrees or π radians.

For instance, if two angles, α and β, are given, the third angle, γ, can be calculated as:

γ 180° - (α β)

or in radians:

γ π - (α β)

Using the Law of Sines to Find the Remaining Angles

If you know two angles and the length of one side, you can use the Law of Sines (also known as the Sine Rule) to find the lengths of the other sides and the remaining angle. The Law of Sines states:

[ frac{a}{sin A} frac{b}{sin B} frac{c}{sin C} ]

Where (a, b, c) are the sides opposite to angles (A, B, C) respectively.

To find one of the unknown angles, say (B), you can use the following step:

[ sin B frac{b sin A}{a} ]

Then, (B arcsin(sin B)).

The third angle, (C), can then be found as:

C 180° - A - B (or in radians: C π - A - B)

Using the Law of Cosines for Sides

When you have one side and the two angles, you can use the Law of Cosines to find the other sides of the triangle. The Law of Cosines states:

[ c^2 a^2 b^2 - 2abcos C ]

To find angle (A), you can first rearrange the formula to solve for (cos A):

[ cos A frac{b^2 c^2 - a^2}{2bc} ]

Then, (A arccos(cos A)).

After finding angle (A), you can use the Law of Sines to find the remaining angle (B):

[ B arcsinleft(frac{b sin A}{a}right) ]

Finally, the third angle, (C), is:

C 180° - A - B (or in radians: C π - A - B)

Conclusion

When working with triangles, the methods for finding angles and sides depend on the given information. Whether you use the simpler sum of angles formula, the Law of Sines, or the Law of Cosines, these tools from trigonometry can help you solve complex geometric problems.