A triangle has sides 5 and 8 with an included angle of 60 degrees. How long is the third side?

1. Two sides with the angle between them is the SAS case, so use the Law of Cosines. 2. The unknown side c satisfies c squared equals a squared plus b squared minus 2ab cos of the angle. 3. Here a squared is 25 and b squared is 64, so their sum is 89. 4. The cosine of 60 degrees is 0.5. 5. So 2ab cos of the angle is 2 times 5 times 8 times 0.5, which is 40. 6. Subtracting gives c squared equal to 89 minus 40, which is 49. 7. Taking the square root gives c equal to 7 units. 8. Answering 9 units drops the cosine term and takes the square root of 89 loosely. 9. Answering 13 units simply adds the two sides, which only holds for a flat straight line. 10. Answering 11 units adds the 2ab term instead of subtracting it. _Source: OpenStax Precalculus (CC BY 4.0), Ch 8 "Further Applications of Trigonometry", section 8.2 Non-right Triangles: Law of Cosines_