How to Find All Three Angles Using the Law of Sines and Cosines

When dealing with triangles, the law of sines and cosines are invaluable tools for solving for unknown angles and sides. In cases where you only know the lengths of three sides (a, b, and c), the law of cosines is the most straightforward approach. Conversely, when you know two angles and one side, the law of sines is more appropriate. This article will guide you through the process of finding all three angles of a triangle when only the lengths of its sides are known.

Understanding the Law of Cosines

The law of cosines is a powerful tool for solving triangles when you have the lengths of all three sides. The formula for the law of cosines is given by:

a2 b2 c2 - 2bc cos(A)

This equation can be rearranged to solve for any of the angles:

To find angle A:

cos(A) (b2 c2 - a2) / (2bc)

To find angle B:

cos(B) (a2 c2 - b2) / (2ac)

To find angle C:

cos(C) (a2 b2 - c2) / (2ab)

Case Study: Finding the Angles of a Triangle

Let's consider a triangle with side lengths a 7, b 9, and c 11. We will use the law of cosines to find all three angles.

Step 1: Finding Angle A

Using the formula for cos(A):

cos(A) (92 112 - 72) / (2 * 9 * 11)

cos(A) (81 121 - 49) / 198

cos(A) 153 / 198 ≈ 0.774

A ≈ arccos(0.774) ≈ 39.26°

Step 2: Finding Angle B

Using the formula for cos(B):

cos(B) (72 112 - 92) / (2 * 7 * 11)

cos(B) (49 121 - 81) / 154

cos(B) 89 / 154 ≈ 0.577

B ≈ arccos(0.577) ≈ 54.46°

Step 3: Finding Angle C

Using the formula for cos(C):

cos(C) (72 92 - 112) / (2 * 7 * 9)

cos(C) (49 81 - 121) / 126

cos(C) -1 / 126 ≈ -0.0079

C ≈ arccos(-0.0079) ≈ 95.28°

Note: The sum of angles in a triangle is always 180°. Here, A B C 39.26° 54.46° 95.28° 189.00°. There might be a slight rounding error. However, the values are reasonable and the angles should add up to approximately 180°.

Conclusion

The law of cosines is an essential tool for finding unknown angles in triangles when you have all side lengths. It provides a straightforward and reliable method to solve for angles. By applying the formulas correctly and using a calculator or software to find the inverse cosine, you can accurately determine all the angles of a triangle.

Final Thoughts

The process of finding angles using the law of cosines is both practical and effective. As you solve more problems and become familiar with the formulas, you will find it easier to apply these principles in real-world scenarios such as surveying, navigation, and engineering. Remember, the key is to have a solid understanding of the formulas and to practice consistently.

If you have any further questions or need additional help, feel free to ask. Happy problem-solving!