Comet tapes a paper unit circle to the Boathouse wall and marks every 15 degrees with chalk.
"arcsin lives from -90° to 90°," Wren says. "What about arccos? Cosine is the x-coordinate. Where does it only fall?"
"From 0° up to 180°," Comet says, tracing the top half. "Cosine goes from 1 down to -1 and never repeats."
"And tangent rises the whole way from -90° to 90°," Wren says. "Same range as arcsin, but the ends are not allowed."
Nova projects a horizontal line sliding up the sine wave. "Would you like a hint? Count the crossings in one turn."
"Two," Comet says. "The inverse gives me one. The circle gives me the other, straight across the axis."
"Two ways to find the second angle," Wren says. "Let us compare them."
| Function | Restricted domain | Inverse | Range of the inverse |
|---|---|---|---|
| sin x | -90° to 90° | arcsin | -90° to 90° |
| cos x | 0° to 180° | arccos | 0° to 180° |
| tan x | between -90° and 90° | arctan | between -90° and 90° |
Each range is a piece where the function only rises or only falls. That is exactly what a restricted domain is for.
Tangent at 90° has no value, so arctan never gives 90° or -90°. Its answers stay strictly between them.
Examples: arcsin(√3/2) = 60°, arccos(-√2/2) = 135°, arctan(-√3) = -60°.
The inverse gives the principal value. For sin t = 1/2 that is 30°, a point in the first quadrant.
Sine is the y-coordinate, so the mirror point across the y-axis has the same sine. That angle is 180° - 30° = 150°.
For cosine, mirror across the x-axis: cos t = -1/2 at 120° and at 360° - 120° = 240°.
For tangent, go straight through the center: tan t = 1 at 45° and at 45° + 180° = 225°.
Draw y = sin t from 0° to 360° and the line y = 1/2. The line crosses the wave twice, at 30° and 150°.
The graph shows at a glance how many solutions one turn holds. The circle tells you exactly where they are.
The crew keeps both in the log. If the two ways disagree, a sign slipped somewhere.
| Equation | Principal value | Second angle in one turn | Rule used |
|---|---|---|---|
| sin t = 1/2 | 30° | 150° | 180° - angle |
| cos t = -1/2 | 120° | 240° | 360° - angle |
| tan t = 1 | 45° | 225° | angle + 180° |
| sin t = -√2/2 | -45° | 225° and 315° | add 360°, then 180° - angle |
When arcsin gives a negative angle, add 360° to land in one turn. Then mirror as usual.
| Statement | True or false? |
|---|---|
| arccos(x) is always between 0° and 180°. | ? |
| arctan(x) can equal 90°. | ? |
| If sin t = 1/2, the two solutions in one turn are 30° and 150°. | ? |
| If cos t = -1/2, the second solution is 180° - 120° = 60°. | ? |
| Solutions of tan t = k in one turn are 180° apart. | ? |
Excellent. Tomorrow the protractor and a cardboard circle turn the gauge wave into your own table of readings.