Identity Puzzles

Pre-Calculus · Axiom Academy

Some equations are true for one angle. A few are true for every angle — and those are the ones that hand you superpowers. An equation that refuses to break Solve x + 3 = 7 and you get exactly one answer: x = 4 . Plug in anything else and it's false. Most equations are like that — picky, true for only a special value. But a handful of trig equations are true no matter what angle you feed them. Those special ones are called identities , and they're the engine behind everything you'll do with trigonometry. Watch the angle sweep all the way around the circle. The blue bar is ^2 and the teal bar is ^2 , stacked together. The pieces trade size as turns — but their total stays pinned at exactly 1 , the whole way around. The bars swap shares as the angle turns, but the stack never grows past — or falls short of — exactly 1. That stubbornness is what makes ^2 + ^2 = 1 an identity. Hunt for an angle that breaks it Grab the dial and send anywhere you like — a nice angle, an ugly one, negative, past a full turn. The readout computes the real ^2 and ^2 and adds them. Try as hard as you want to make the total miss 1 . No angle breaks it. An equation that's true for every value of the variable is exactly what we mean by an . An impostor that's only sometimes true Here's a look-alike: + = 1 . Drag and watch the dot, which sits at the real value of + , against the dashed target line at 1 . Sometimes the dot lands right on the line. Most of the time it misses — sometimes wildly.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.