A triangle has sides 7, 8 and 13. What is the angle opposite the side of length 13?

1. All three sides are known and no angle is, so this is the SSS case. 2. Rearranged, the cosine of the angle equals a squared plus b squared minus c squared, all over 2ab. 3. Here the two shorter sides are 7 and 8, and the opposite side is 13. 4. The numerator is 49 plus 64 minus 169, which is negative 56. 5. The denominator is 2 times 7 times 8, which is 112. 6. So the cosine of the angle is negative 56 over 112, which is negative 0.5. 7. The angle whose cosine is negative 0.5 is 120 degrees. 8. Answering 60 degrees ignores the minus sign and takes the cosine as positive 0.5. 9. Answering 90 degrees would need the numerator to vanish, so 49 plus 64 would have to equal 169. 10. Answering 150 degrees would need a cosine near negative 0.87, not negative 0.5. _Source: OpenStax Precalculus (CC BY 4.0), Ch 8 "Further Applications of Trigonometry", section 8.2 Non-right Triangles: Law of Cosines_