October 7, 2025

What Is The Law Of Cosines

Q: How is the Law of Cosines related to the Pythagorean theorem?

The Pythagorean theorem is a special case of the Law of Cosines. If the angle C is a right angle (90°), its cosine is 0. The formula c² = a² + b² - 2ab cos(C) simplifies to c² = a² + b², which is the Pythagorean theorem.

Q: When should I use the Law of Sines instead of the Law of Cosines?

Use the Law of Sines when you know two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA). Use the Law of Cosines when you know two sides and their included angle (SAS) or all three sides (SSS).

Q: Can the Law of Cosines result in more than one possible triangle?

No, unlike the Law of Sines which can have an 'ambiguous case' (SSA), the Law of Cosines provides a unique solution. When used in SAS or SSS scenarios, it defines a single, specific triangle.

Q: What happens if I calculate the cosine of an angle and get a negative number?

If solving for an angle gives you a negative cosine value, it simply means that the angle is obtuse (greater than 90° and less than 180°). This is a normal and expected result for triangles with an obtuse angle.

Learn the Law of Cosines, a key trigonometric formula used to find missing sides or angles in any triangle. Includes the formula, examples, and key applications.

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Understanding the Law of Cosines

The Law of Cosines is a fundamental theorem in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is a generalization of the Pythagorean theorem and can be applied to any type of triangle, not just right triangles.

The Law of Cosines Formula

For any triangle with sides a, b, and c, and opposite angles A, B, and C, the Law of Cosines is expressed in three forms: c² = a² + b² - 2ab cos(C); a² = b² + c² - 2bc cos(A); and b² = a² + c² - 2ac cos(B). Each formula allows you to find a missing side if you know the other two sides and their included angle.

A Practical Example: Finding a Side

Imagine a triangle where side a = 8, side b = 10, and the angle between them, C, is 60°. To find the length of side c, you would use the formula c² = a² + b² - 2ab cos(C). Plugging in the values gives c² = 8² + 10² - 2(8)(10)cos(60°), which simplifies to c² = 64 + 100 - 160(0.5) = 84. Therefore, side c is the square root of 84, approximately 9.17.

Key Applications

The Law of Cosines is essential in two main scenarios: 1) when you know the lengths of two sides and the measure of the included angle (Side-Angle-Side, or SAS) and need to find the third side. 2) when you know the lengths of all three sides (Side-Side-Side, or SSS) and need to find the measure of any angle.

Frequently Asked Questions

How is the Law of Cosines related to the Pythagorean theorem?

When should I use the Law of Sines instead of the Law of Cosines?

Can the Law of Cosines result in more than one possible triangle?

What happens if I calculate the cosine of an angle and get a negative number?