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Precalculus 9-12 / Week 08 / Dock Day
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Week 08 · Laws of Sines and Cosines

Dock Day

Dock Day: across the puddle
// The buoy sighted from shore
⏱ about 20 min

Dock Day: Across the Puddle

Saturday the tape, the protractor and a ball of string come home in Comet's pack.

"Find something you cannot walk across," she says. "A puddle, a flowerbed, a hedge. Then measure to it anyway."

"Lay a baseline along one side," Wren says. "Sight the far point from both ends. Two angles and a tape."

Nova dims her light. "Would you like a hint? The third angle comes free."

"Then the Law of Sines gives both distances," someone says, "and a height gives the straight gap."

"Measure around to check if you can," Comet says. "If you cannot, the triangle is the only way, and that is the point."

  1. An oblique triangle has no right angle. Drop a height and you make two right triangles that share it.
  2. Law of Sines: a/sin A = b/sin B = c/sin C. Use it for ASA and AAS, once a side has its own angle.
  3. Law of Cosines: c² = a² + b² - 2ab cos C. Use it for SAS, and solved for cos C for SSS.
  4. Area = ½ab sin C, with C the included angle. With C = 90° the cosine law is the Pythagorean theorem.
  5. Find a possibly obtuse angle with cosines first. Check that the biggest side faces the biggest angle.
◇ FAMILY BASELINE
Pick a far point across something you cannot walk through. Lay a baseline of string along your side of it.
One person sights the far point from each end with the protractor flat on the string. Record both angles.
Together, find the third angle, then use the Law of Sines for the two distances. Then find the straight gap as a height.
If there is a way around, tape the straight gap and compare. Trade roles and try a second target.
For a grown-up
Stay on dry, flat ground. Nobody steps into the puddle, bed or hedge to check; the triangle does the checking.
A protractor read to the nearest degree is plenty. Expect the computed and taped gaps to differ a little.
All distances here are the family's own measurements, not facts about any real place.
DOCK DAY CHECK
  • Read the question.
  • Tap your answer.
A triangle ABC with angles A = 70°, B = 60°, C = 50° and sides a = 6, b = ?A family baseline faces a 70° angle at the far point and is 6 m long. Find the side facing the 60° angle, to 1 place.
A triangle ABC with angles A = ?, B = ?, C = 50° and sides a = 6, b = 5A family triangle has sides 6 m and 5 m with a 50° angle between them. What is its area, to 1 place?
A family baseline has angles of 50° and 60° at its ends. What is the angle at the far point? Type the number.
WHY THIS EXERCISEThe third angle is the one across from the baseline, and that pair starts the Law of Sines every time.
Draw your baseline, the far point and both sight lines. Label the two angles you measured and the gap you found.
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