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Precalculus 9-12 / Week 07 / Dock Day
6/6
Week 07 · Inverse Trig and the Water Wave

Dock Day

Dock Day: the family wave
// Reading the gauge backwards
⏱ about 20 min

Dock Day: The Family Wave

Saturday the cardboard wheel comes home with a ruler, a protractor and a stack of index cards.

"Spin it to a secret angle," Comet says. "Measure the mark's height and call out the number. We guess the angle."

"There are two right answers every time," Wren warns. "Unless the mark is at the very top or bottom."

Nova dims her light. "Would you like a hint? Divide the height by the radius first."

"Height 5, radius 10, sine one half," comes the answer. "So 30° or 150°." The wheel shows 150°.

"Now backwards with the clock," Comet says. "The tip of the minute hand rises and falls like the gauge."

"Pick a height," Wren says. "Find both minutes in the hour it happens."

  1. Sine repeats, so on its whole domain it has no inverse. Restricted to -90° to 90°, it only rises, and arcsin exists.
  2. arccos lives on 0° to 180°, arctan strictly between -90° and 90°. Each gives one angle, the principal value.
  3. The unit circle finds the other solutions: sine mirrors across the y-axis, cosine across the x-axis, tangent through the center.
  4. To solve a sin(bt) + d = k: isolate the sine, take arcsin, mirror, add 360° per cycle, divide by b.
  5. Read every answer in the gauge's units, and say when a solution falls outside the day or never happens.
◇ FAMILY WHEEL AND CLOCK
Spin the cardboard wheel to a secret angle. Measure the mark's height above the center and call it out.
Everyone else divides by the radius, finds the principal angle, and names the mirror angle. Check the wheel.
Trade roles so everyone spins and everyone solves. Keep a tally of first-try matches.
Then watch a clock with hands. Pick a height for the minute hand's tip and find both minutes in the hour it is reached.
For a grown-up
The wheel is cardboard on a pin. The grown-up sets the pin and takes it out afterward.
A wrong answer is a good moment: measure again and ask whether the mark is above or below the center line.
The wheel and the gauge wave are the crew's own game, not a fact about any real lake or clock.
DOCK DAY CHECK
  • Read the question.
  • Tap your answer.
The wheel mark is 5 centimeters above center on a 10 centimeter wheel. What is arcsin(1/2)?
The mark is about 8.7 centimeters up on a 10 centimeter wheel, so sin t = √3/2. Which angles under 360° work?
The gauge model is 1.2 sin(30t) + 3.4. The water first reads 4.6 meters, the peak, after how many hours? Type the number.
WHY THIS EXERCISEThe peak is the one reading per cycle with a single solution, the top of the wheel.
Draw the clock face with the minute hand at the two times its tip is at your chosen height.
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