Tuesday

Wren chalks two axes through the center of the circle, so the stand sits at the origin.

"Swing the string to 30° and mark where it meets the circle," he says. Comet marks it and measures.

"About 0.87 across and 0.5 up," she reads. "Those look like my 30-60-90 triangle."

"They are," Wren says. "The hypotenuse is the string, length 1. So the legs are cos 30° and sin 30°."

Nova lights the point. "Would you like a hint? Every point on this circle is (cos θ, sin θ)."

"Even past 90°?" Comet asks, swinging the string to 150°. "The x is negative now."

"Then cos 150° is negative," Wren says. "Same size as cos 30°, other sign. The circle decides."

Start at (1, 0) and turn counterclockwise through an angle θ. The point where the arm meets the circle is (cos θ, sin θ).

For an acute angle, this is the right-triangle definition from Geometry, with hypotenuse 1.

For any other angle, it is the definition. Cosine is the x-coordinate, sine is the y-coordinate, for every θ.

A 45-45-90 triangle with hypotenuse 1 has legs √2/2 each. So the 45° point is (√2/2, √2/2).

A 30-60-90 triangle with hypotenuse 1 has legs 1/2 and √3/2. So 30° gives (√3/2, 1/2) and 60° gives (1/2, √3/2).

These exact values are the whole first quadrant. The other quadrants reuse them with signs.

Way one, by symmetry: the 150° point is the mirror of the 30° point across the y-axis. Same y, opposite x.

Way two, by reference angle: 180° - 150° = 30°. Take cos 30° = √3/2, then make it negative because quadrant 2 has negative x.

Both ways give cos 150° = -√3/2 and sin 150° = 1/2. The crew uses the reference angle, and checks with the picture.

Excellent. Tomorrow the tape measure meets the chalk circle in the Glass House Lab.