What is the law of cosines?

• A. The law of cosines relates the sides to the tangents of the angles
• B. Given the lengths of two sides of a triangle and the included angle, this law will provide the length of the third side
• C. The law of cosines states that a^2 + b^2 = c^2 for a right triangle
• D. The law of cosines computes the area of a triangle

Answer: (B)

Think of a triangle where you know two sides and the angle between them. The law of cosines lets you find the third side with the formula c^2 = a^2 + b^2 − 2ab cos(C). This works for any triangle, not just right ones. It also shows the link to the Pythagorean theorem: if the included angle is 90 degrees, cos(90°) = 0, so c^2 = a^2 + b^2.

So, by plugging in the two known sides and the included angle, you solve for the unknown side. For example, with a = 5, b = 7, and C = 60°, the third side is c = sqrt(5^2 + 7^2 − 2·5·7·cos60°) = sqrt(39) ≈ 6.24.

This law is not about tangents, and it isn’t simply the Pythagorean relation unless the angle is right; it also doesn’t directly give the area of a triangle.