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Precalculus 9-12 / Week 07 / Thursday
4/6
Week 07 · Inverse Trig and the Water Wave

Thursday

Every time in a day
// Reading the gauge backwards
⏱ about 20 min

Thursday: Every Time in a Day

"The water hit 4 meters 1 hour after dawn," Wren says. "But a day is 24 hours. When else?"

Comet writes sin(30t) = 0.5 and circles 30° and 150°. "Divide by 30: t = 1 and t = 5."

"Then the wave repeats every 12 hours," Wren says. "So add 12: 13 and 17."

"Four times in a day," Comet says. "What do you notice? They come in pairs, 4 hours apart."

Nova projects a second wave, lower and flatter. "Would you like a hint? The small bay has its own post."

"h = 0.9 sin(30t) + 2.8," Wren reads. "When is it 2.5? That sine will not be a nice fraction."

"Then we use the arcsin key," Comet says. "And we still finish with the circle."

Solving a sin(bt) + d = k, all the way

  1. Subtract d, then divide by a, so the sine is alone: sin(bt) = (k - d)/a.
  2. Check the number is between -1 and 1. If not, the reading never happens.
  3. Take arcsin for the principal angle. Use the exact table if you can, the calculator if not.
  4. Find the second angle in one turn: 180° minus the principal angle.
  5. Add 360° to each angle for every extra cycle you need.
  6. Divide every angle by b to turn it back into t, and read each t in the gauge's units.

For the main post: angles 30°, 150°, 390°, 510°. Divide by 30: t = 1, 5, 13, 17 hours.

For the small bay: sin(30t) = (2.5 - 2.8)/0.9 = -0.333. Arcsin gives -19.5°, a negative angle.

Add 360° to get into one turn, then mirror for the second angle. Divide by 30: t ≈ 6.6 and 11.4 hours.

Why sine needs a restricted domain

Here is the argument behind the whole week, in six steps. Read it first, then put it in order below.

  1. A function pairs each input with exactly one output.
  2. An inverse swaps inputs and outputs, so each old output must come from exactly one old input.
  3. Sine repeats: sin 30° = sin 150° = sin 390°. One output, many inputs.
  4. So sine on its whole domain cannot have an inverse.
  5. From -90° to 90°, sine only rises, so every output appears exactly once.
  6. On that restricted domain the inverse exists. We call it arcsin.
THE ARGUMENT, IN ORDER
  • ?A function gives each input exactly one output
  • ?An inverse needs each output to come from exactly one input
  • ?Sine repeats: sin 30° = sin 150° = sin 390°
  • ?So sine on its whole domain has no inverse
  • ?From -90° to 90° sine only rises, so no output repeats
  • ?On that restricted domain the inverse, arcsin, exists
WHY THIS EXERCISEA derivation is a chain. The restricted domain is not a trick; it is the smallest fix that makes an inverse possible.
A function must be always rising or always falling on its domain to have an inverse. We shrink the domain to make that true. This smaller domain is called a this domain. Type one word.
The one angle an inverse trig function returns is called the this value. Type one word.
The gauge wave repeats every 12 hours. That number is the wave's this. Type one word.
SOLVE THE GAUGE MODEL
  • Read the question.
  • Tap your answer.
The crew's gauge model is h = 1.2 sin(30t) + 3.4. When is h first equal to 4? (t in hours, round to 1 place.)
The crew's gauge model is h = 1.2 sin(30t) + 3.4. When is h first equal to 2.8? (t in hours, round to 1 place.)
The crew's gauge model is h = 0.9 sin(30t) + 2.8. When is h first equal to 2.5? (t in hours, round to 1 place.)
The main post reads 4 meters at t = 1 and t = 5. The period is 12 hours. What is the third time in the day?
How many times in 24 hours does the main post read 4 meters? Type the number.
WHY THIS EXERCISECounting solutions is reading the wave: two crossings per period, times the number of periods.

Which solution makes sense?

Dividing 510° by 30 gives 17, a real solution. Dividing -30° by 30 gives -1, an hour before dawn.

A negative t is good algebra and still outside the day the crew logged. Say so, and move on.

Setting h = 5 gives sin(30t) = 1.33, more than 1. The main post never reads 5 meters in this model.

StatementTrue or false?
Every solution of a sin(bt) + d = k is found by dividing the angles by b.?
The main post reads 5 meters at some time in the crew's model.?
The second angle in one turn is 180° minus the principal angle, for sine.?
1 + 12 = 13?
A negative t means the model is wrong.?
WHY THIS EXERCISESolving is half the job. Reading each answer in the gauge's units is the other half.

Clear reasoning. Tomorrow inverse trig shows up all around the Boathouse, and the week comes back in a mixed set.

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