# Angle patterns on the unit circle

The student blinked, and the Mathematical Ninja had covered the board in a colourful diagram.

“It’s easy,” said the Mathematical Ninja, “to work out the fractions of pi in each quadrant - once you know a few rules.”

The student automatically reached for a pen. Whenever the Mathematical Ninja mentioned rules, good stuff was coming; more to the point, if the Mathematical Ninja mentioned rules and you didn’t reach for a pen, you’d be dangling out of the window by your angles within seconds. He’d tried that once. It wasn’t fun.

“There’s a pattern,” said the Mathematical Ninja, “in the bottoms. The shallow lines in green all have 6 on the bottom; the moderate lines in red all have four on the bottom and the steep lines in blue all have 3 on the bottom.”

“And the vertical line has two on the bottom?” ventured the student.

“Correct! And the tops, there’s a pattern for those as well. In the first quadrant, top right, it’s all ones. Dead easy. In the second quadrant, top left, the top is one less than the bottom. Bottom left, third quadrant, it’s one more than the bottom. And in the bottom right…”

“It’s one less than double the bottom,” smiled the student, for once a step ahead of the Mathematical Ninja.

The Mathematical Ninja bowed slightly, with a smirk on his face. “You have learned well,” he said. “There will be a short test next week.”