How to Solve Triangles Using the Law of Cosines

TL;DR

To solve triangles with the Law of Cosines, use the formula c² = a² + b² - 2ab cos(C) when you have two sides and the included angle or all three sides. If the triangle has different letters for sides and angles, the Law of Sines is not applicable. After finding one angle using the Law of Cosines, the Law of Sines can help you calculate the other angles.

Transcript

let's say side a is 10 and side b is 20 and angle c is 60 degrees go ahead and solve the triangle so first let's draw it so this is angle a b and c so angle c is 60 degrees side a is 10 side b is 20. so what we have is a side angle side triangle can we use law of sines to solve the triangle in order to use law of sines you need to have two of the s... Read More

Key Insights

• 💁 The Law of Sines and Law of Cosines are essential tools in solving triangles, depending on the given information.
• 🙃 The Law of Sines requires two sides and their corresponding angles, while the Law of Cosines is used when two sides and their included angle are known.
• 🙃 The Law of Cosines can be rearranged into three different forms to solve for different sides and angles.

Questions & Answers

Q: When can the Law of Sines be used to solve a triangle?

The Law of Sines can be used when the triangle has two sides and their corresponding angles known, or when it has two angles and one side known. It cannot be used if all the letters (sides and angles) are different.

Q: How do you use the Law of Cosines to solve a triangle?

To use the Law of Cosines, you need to know the lengths of two sides and the measure of the angle between them. Plug these values into the formula c² = a² + b² - 2ab cos(C) to solve for the remaining side or angle.

Q: Can the Law of Sines be used to solve a right triangle?

Yes, the Law of Sines can be used to solve a right triangle. However, it is more efficient to use the Pythagorean theorem to find the missing side length.

Q: In what instances can the Law of Cosines be used to solve a triangle?

The Law of Cosines can be used to solve a triangle when you have three sides, two sides and the included angle, or one side and two angles known. This formula can handle cases when all letters (sides and angles) are different.

Summary & Key Takeaways

• The Law of Sines and Law of Cosines are used to solve triangles when given different combinations of side lengths and angles.

• If a triangle has all different letters (sides and angles), the Law of Sines cannot be used, and the Law of Cosines should be applied.

• The formula for the Law of Cosines is c² = a² + b² - 2ab cos(C), which can be rearranged into three different forms to solve for different sides and angles.

• When given all three side lengths, the Law of Cosines is used to find one angle, and then the Law of Sines can be used to find the remaining angles.