1. Use the Law of Cosines: c² = a² + b² - 2ab cos(C). Here, a = 5 m, b = 7 m, C = 15°.
c² = 5² + 7² - 2(5)(7)cos(15°)
c² = 25 + 49 - 70 cos(15°)
c² = 74 - 70(0.9659) ≈ 74 - 67.613 ≈ 6.387
c ≈ √6.387 ≈ 2.53 meters.
The skaters are approximately 2.53 meters apart.
2. Use the Law of Cosines: cos(A) = (b² + c² - a²)/(2bc). Here, a = 46 ft, b = 63 ft, c = 40 ft.
cos(A) = (63² + 40² - 46²)/(2×63×40)
cos(A) = (3969 + 1600 - 2116)/5040
cos(A) = (3453)/5040 ≈ 0.6851
A ≈ cos⁻¹(0.6851) ≈ 46.8°.
The angle opposite the 46-foot side is approximately 46.8°.
3. The distance between A and B is ½ mile = 0.5 miles. The angles at A and B are 63° and 56°, so angle C = 180° - 63° - 56° = 61°.
Use the Law of Sines: a/sin(A) = b/sin(B) = c/sin(C).
Let a = distance from B to C, b = distance from A to C, c = 0.5 miles.
b/sin(56°) = 0.5/sin(61°)
b = (0.5 × sin(56°))/sin(61°) ≈ (0.5 × 0.8290)/0.8746 ≈ 0.4145/0.8746 ≈ 0.474 miles.
a/sin(63°) = 0.5/sin(61°)
a = (0.5 × sin(63°))/sin(61°) ≈ (0.5 × 0.8910)/0.8746 ≈ 0.4455/0.8746 ≈ 0.510 miles.
The distance from observer A to the sailboat is approximately 0.474 miles, and from observer B is approximately 0.510 miles.
4. Let h be the height of the flagpole. The man is 400 feet from the base of the building.
Let H be the height of the building. The angle to the bottom of the flagpole is 60°, so tan(60°) = H/400 → H = 400 tan(60°) = 400√3 ≈ 692.82 feet.
The angle to the top of the flagpole is 62.5°, so tan(62.5°) = (H + h)/400 → H + h = 400 tan(62.5°).
tan(62.5°) ≈ 1.920, so H + h ≈ 400 × 1.920 = 768.0 feet.
h = 768.0 - H ≈ 768.0 - 692.82 = 75.18 feet.
The height of the flagpole is approximately 75.2 feet.
Parent Tip: Review the logic above to help your child master the concept of law of sines cosines worksheet.
