Trigonometry · Quick Reference

Law of Cosines Cheat Sheet

Trigonometry · Lesson 7/8

In one line: the Law of Cosines generalizes the Pythagorean theorem to work with ANY triangle, not just right triangles — essential when you know sides but not a convenient right angle.

Key Ideas

1The Law of Cosines Formula. c^2 = a^2 + b^2 - 2ab*cos(C), where C is the angle opposite side c.

2Relation to the Pythagorean Theorem. If C = 90°, cos(C) = 0, and the formula reduces exactly to the Pythagorean theorem: c^2 = a^2 + b^2.

3When to Use It. Use the Law of Cosines when you know: all three sides (SSS), or two sides and the included angle (SAS).

4Finding a Missing Side. Plug the two known sides and included angle directly into the formula to solve for the third side.

5Finding a Missing Angle. Rearrange the formula to solve for cos(C), then take the inverse cosine to find the angle.

Worked Examples

A triangle has sides a=7, b=10, and included angle C=60°. Find side c.

c^2=79, so c is approximately 8.89

A triangle has sides a=5, b=6, c=7. Find angle C (opposite side c) using the Law of Cosines.

cos(C)=0.2, so C = arccos(0.2), approximately 78.5°

Show that the Law of Cosines matches the Pythagorean theorem when C=90°.

c^2 = a^2+b^2, exactly the Pythagorean theorem

Formulas

c^2 = a^2 + b^2 - 2ab*cos(C)

cos(C) = (a^2+b^2-c^2) / (2ab)

Practice Yourself

A triangle has a=6, b=8, C=90°. Find c using the Law of Cosines.

c=10 (matches the Pythagorean theorem)

A triangle has a=4, b=5, C=60°. Find c^2.

c^2=16+25-2*4*5*0.5=21

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

When you know SSS or SAS (not a matching angle-side pair)