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Precalculus 9-12 / Week 07 / Friday
5/6
Week 07 · Inverse Trig and the Water Wave

Friday

Inverses around the Boathouse
// Reading the gauge backwards
⏱ about 20 min

Friday: Inverses Around the Boathouse

Friday the Boathouse wall is covered in angles found backwards.

"The gauge hit 4 at 1, 5, 13, 17 hours," Comet reads. "The small bay hit 2.5 at 6.6 and 11.4."

"And the ramp," Wren says. "It rises 1 meter over a 4 meter run. What angle is that? arctan of one quarter."

"The winch rope leaves the drum at some angle too," Comet says. "Rope length, height, arcsin."

Nova sorts the wall into two columns. "Would you like a hint? One column asks for a number, the other for an angle."

"Sine, cosine, tangent give numbers," Wren says. "Their inverses give angles. Same facts, read the other way."

"Then let us read the whole week the other way," Comet says.

Where the crew used inverses this week

QuestionEquationInverse stepAnswer
When does the gauge first read 4 m?1.2 sin(30t) + 3.4 = 430t = arcsin(0.5) = 30°t = 1 hour
Every time in a day it reads 4 msin(30t) = 0.530°, 150°, then add 360°t = 1, 5, 13, 17
When does the small bay read 2.5 m?0.9 sin(30t) + 2.8 = 2.5arcsin(-0.333), then mirrort ≈ 6.6, 11.4
Which angle has cosine -1/2?cos t = -1/2arccos(-1/2) = 120°120° and 240° in one turn

Every row has the same shape: isolate the trig function, take the inverse, then use symmetry for the rest.

When the ratio is in the exact table, you can name the angle. Otherwise the arcsin key gives a decimal.

Everyday angles found backwards

  • A ramp's slope is a tangent. Its angle is arctan(rise ÷ run), and the crew's ramp rises 1 over 4.
  • A ladder against a wall makes an angle whose cosine is base ÷ ladder length. A grown-up handles the ladder.
  • A wheel's valve rises and falls like the gauge. Any height it reaches, it reaches twice per turn.
  • A swing seat's height above its lowest point repeats each swing, two matching times per period.

Which answer makes sense?

A calculator says arcsin(0.25) ≈ 14.5°. The ramp question wants arctan, not arcsin. Pick the inverse that matches the ratio you have.

The wheel question "what angle has height one half" has two answers in a turn, 30° and 150°. The inverse alone gives only the first.

MIXED SET
  • Read the question.
  • Tap your answer.
A ramp rises √3 meters for each meter of run in the crew's sketch. What angle does it make, arctan(√3)?
A ladder's base ÷ length is √2/2 in the crew's drawing. What is the angle at the ground, arccos(√2/2)?
Find every angle t from 0° up to 360° where the wheel's x-coordinate, cos t, equals 1/2.
The crew's gauge model is h = 1.2 sin(30t) + 3.4. When is h first equal to 3.7? (t in hours, round to 1 place.)
arcsin(1/2) = ? Type the number of degrees.
arccos(-1) = ? Type the number of degrees.
The gauge model is 1.2 sin(30t) + 3.4. When does it first read 2.8? Type the number of hours.
Week reviewTrue or false?
Restricting sine to -90° to 90° lets it have an inverse.?
arccos gives angles from -90° to 90°.?
A ramp's angle is arctan(rise ÷ run).?
The main post reads 4 meters 4 times in 24 hours in the crew's model.?
sin t = 1.4 has two solutions in one turn.?
WHY THIS EXERCISEThe week in five lines: restrict, invert, mirror, repeat, and read the answer in its units.
In the crew's model, how many hours apart are the first two times the gauge reads 4? Type the number.
WHY THIS EXERCISEThe gap between paired solutions is a reading you can check against the real gauge tomorrow.
Draw the ramp, the ladder and the gauge post. Label the inverse function that finds each angle.

A full week of angles found backwards. Tomorrow is Dock Day, and the family measures a wave at home.

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