NY Regents Geometry Law of Cosines — practice questions

1. The Law of Sines pairs an angle with the side opposite it, so it needs such a pair. 2. With two sides and the angle between them, no angle has its opposite side known. 3. With three sides and no angle at all, there is no pair to start from either. 4. Those two cases, SAS and SSS, are exactly what the Law of Cosines handles. 5. ASA and AAS both hand over an angle with its opposite side, so the Law of Sines suffices. _Source: OpenStax Precalculus (CC BY 4.0), Ch 8 "Further Applications of Trigonometry", section 8.2 Non-right Triangles: Law of Cosines_

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_

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_