Monday

Dawn mist sits on the lake. Wren wades to the depth gauge post in his life vest while a grown-up watches from the dock.

"3.4 meters," he calls. "Same as yesterday at this hour. Nova, what does the log say for the whole day?"

Nova projects a wave over the water. "The crew's readings rise to 4.6, fall to 2.2, and repeat every 12 hours."

"A sine wave," Comet says, sketching it. "h = 1.2 sin(30t) + 3.4. So when does the water first reach 4 meters?"

Wren frowns. "That means sin(30t) = 0.5. I know sin 30° is one half. But sin 150° is one half too."

Nova dims. "Would you like a hint? Which half of the wave are you standing on?"

"The rising half," Comet says, circling one point. "Pick that one first. The rest come from symmetry."

sin 30° = 1/2. So does sin 150°, sin 390° and sin -210°. One output, endless inputs.

A function can only be undone when each output comes from one input. Sine fails that test on its whole domain.

So we restrict sine to -90° to 90°. On that piece it only rises, from -1 up to 1, and never repeats a value.

Restricted sine has an inverse, called arcsin. arcsin(x) is the one angle between -90° and 90° whose sine is x.

Notice the order: undo the add, undo the multiply, then undo the sine. Each step peels one layer off t.

Every number here is the crew's own reading from Nova's log, not a fact about any real lake.

Strong start. Tomorrow arccos and arctan get their own ranges, and the unit circle finds every other solution.