Click image to open full size
Geometry
Law of Sines & Law of Cosines
Law of Sines and Cosines
Related Tools
Related Labs
Related Worksheets
The Law of Sines and the Law of Cosines help you solve triangles that are not right triangles. They connect side lengths and angle measures in any triangle, so they are useful when the Pythagorean theorem or basic sine, cosine, and tangent are not enough. These laws are important in geometry, surveying, navigation, engineering, and map making. They give you a way to find missing parts of a triangle from limited information.
Key Facts
- Law of Sines: a/sin A = b/sin B = c/sin C
- Law of Cosines: c^2 = a^2 + b^2 - 2ab cos C
- Use the Law of Sines when you know ASA, AAS, or sometimes SSA information.
- Use the Law of Cosines when you know SAS or SSS information.
- The angles of any triangle add to 180 degrees: A + B + C = 180 degrees
- In standard triangle notation, side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C.
Vocabulary
- Law of Sines
- A formula that relates each side of a triangle to the sine of its opposite angle.
- Law of Cosines
- A formula that relates the sides of a triangle to the cosine of one included angle.
- Opposite Side
- The side across from a given angle in a triangle.
- Included Angle
- The angle formed between two known sides of a triangle.
- Oblique Triangle
- A triangle that does not have a 90 degree angle.
Common Mistakes to Avoid
- Matching an angle with the wrong side. In the Law of Sines, each angle must be paired with the side directly across from it.
- Using the Law of Sines for SAS information. SAS usually requires the Law of Cosines because the known angle is between two known sides.
- Forgetting to use the 180 degree angle sum. After finding one or two angles, subtract from 180 degrees to find the missing angle.
- Rounding too early in a multi-step problem. Keep extra decimal places until the final answer so the result stays accurate.
Practice Questions
- 1 In triangle ABC, A = 40 degrees, B = 70 degrees, and a = 12 cm. Find angle C, then find side b using the Law of Sines.
- 2 In triangle ABC, a = 8 cm, b = 11 cm, and C = 60 degrees. Find side c using the Law of Cosines.
- 3 A triangle has two known sides and the angle between them. Explain why the Law of Cosines is a better first choice than the Law of Sines.