Solving Triangles: Finding All Three Angles Given Two Sides and the Included Angle

When faced with a triangle problem where you know two sides and the angle between them, you might wonder if it's possible to find the remaining angles. This article delves into the intricacies of such problems and provides a detailed guide on how to find all three angles, leveraging essential theorems and principles.

Introduction to Triangle Solutions

Geometry enthusiasts and students often encounter the challenge of determining the angles of a triangle when given certain side lengths and angles. The key to solving such problems lies in understanding different criteria that can fully identify a triangle. Commonly recognized conditions include the Side-Angle-Side (SAS), Side-Side-Side (SSS), Angle-Side-Angle (ASA), and Angle-Angle-Side (AAS) criteria.

Understanding the SAS and Other Criteria

The Side-Angle-Side (SAS) criteria allow you to find all three angles and sides of a triangle when you know two sides and the angle between them. The other criteria, such as SSS, ASA, and AAS, similarly offer the possibility to define a triangle's dimensions. However, the side-side-side (SSA) or angle-side-side (ASS) criteria do not fully identify the triangle, as they may lead to multiple solutions.

Using the Law of Cosines and Sines

To solve for the angles in a triangle under the SAS conditions, you can use the Law of Cosines and the Law of Sines. The Law of Cosines is particularly useful when the included angle is known, providing a direct way to find the third side of the triangle. The formula is given by:

c^2 a^2 b^2 - 2abcos(C)

Once you have the third side, you can use the Law of Sines to find the other angles. The Law of Sines states:

sin(A)/a sin(B)/b sin(C)/c

By substituting the known values, you can solve for the unknown angles. The final angle can be found using the simple sum of angles theorem for a triangle:

A° 180° - C° - B°

Case Studies and Examples

Let's consider a practical example. Assume you have a triangle with sides a 5 units, b 7 units, and the included angle C 60°. To find the third side, use the Law of Cosines:

c^2 5^2 7^2 - 2 * 5 * 7 * cos(60°) c^2 25 49 - 70 * 0.5 c^2 74 - 35 39 c √39 ≈ 6.24 units

Next, use the Law of Sines to find angles A and B:

sin(A)/5 sin(60°)/6.24 sin(A) (5 * sin(60°))/6.24 ≈ 0.65 A sin^(-1)(0.65) ≈ 40.4°

Similarly, find angle B:

sin(B)/7 sin(60°)/6.24 sin(B) (7 * sin(60°))/6.24 ≈ 0.90 B sin^(-1)(0.90) ≈ 64.2°

Finally, verify:

A B C 40.4° 64.2° 60° 164.6° 15.4° (due to rounding) 180°

Conclusion and Final Thoughts

While the SAS criteria allow for a unique solution to find the remaining angles of a triangle, other conditions like SSA or ASS can lead to multiple solutions or no solution at all. Understanding the principles of the Law of Cosines and Sines is crucial for solving such triangle problems accurately.

Remember, geometry is a field filled with elegance and consistency. By adhering to these principles, you can confidently tackle any triangle-related problems.